Rune Haugseng

Fabian Hebestreit

AND Sil Linskens

Joost Nuiten

LAX MONOIDAL ADJUNCTIONS, TWO-VARIABLE FIBRATIONS AND THE CALCULUS OF MATES

We provide a calculus of mates for functors to the ∞-category of ∞-categories. As the most important application we show that given an adjunction between symmetric monoidal ∞-categories, there is an equivalence between lax symmetric monoidal structures on the right adjoint and oplax symmetric monoidal structures on the left adjoint functor.

As the technical heart of the paper we study various new types of fibrations over a product of two ∞-categories. In particular, we show how they can be dualised over one of the two factors and how they relate to functors out of the Gray tensor product of (∞, 2)-categories.

One of the main goals of the present paper is to prove that if R : C → D is a lax monoidal right adjoint functor of ∞-categories, then its left adjoint has a canonical oplax monoidal structure, and vice versa. More precisely, we will provide a proof of the following statement: Proposition A. Given two symmetric monoidal ∞-categories C and D, the extraction of adjoints gives an equivalence between the ∞-category of lax (symmetric) monoidal right adjoint functors C → D and the opposite of the ∞-category of oplax monoidal left adjoint functors D → C. While Lurie proves in [START_REF]Higher Algebra[END_REF]] that the right adjoint of a strong monoidal functor is lax monoidal, which suffices for a great many applications, the general result has not so far appeared in the literature. The key difference between these two cases is that if the left adjoint is not strong monoidal, we are forced to consider two different descriptions of the symmetric monoidal ∞categories involved. To explain this, let us first recall the definitions:

A symmetric monoidal structure on an ∞-category C can be encoded as a commutative monoid in the ∞-category of ∞-categories (which we will denote simply by Cat) with underlying object C. Recall that a commutative monoid in an ∞-category X with finite products is defined as a functor M : Fin * → X, where Fin * is the category of finite pointed sets, such that the Segal maps M (S + ) → s∈S M (1 + ) are equivalences; here the underlying object is M (1 + ). A homomorphism between such monoids is then simply a natural transformation of functors Fin * → X -in the case of symmetric monoidal ∞-categories these correspond to strong symmetric monoidal functors. If we want to extend this picture to encode (op)lax symmetric monoidal functors, we can either consider the (∞, 2)-categorical notions of (op)lax natural transformations, or we can (equivalently, as we will show) work fibrationally: By the unstraightening theorem a symmetric monoidal ∞-category can be described either as a cocartesian fibration C ⊗ → Fin * , or as a cartesian fibration C ⊗ → Fin op * .

Following Lurie [START_REF]Higher Algebra[END_REF]], a lax monoidal functor F : C → D is then defined as a commutative triangle

C ⊗ D ⊗ Fin * F
where the horizontal functor preserves cocartesian morphisms over inert (but not all) morphisms in Fin * . The lax monoidal structure on F is encoded by the images of cocartesian morphisms over active morphisms in Fin * : for example, the image of the cocartesian arrow (X, Y ) -→ X ⊗ Y in C ⊗ factors into a cocartesian arrow (F (X), F (Y )) → F (X) ⊗ F (Y ), followed by a map in D of the form F (X) ⊗ F (Y ) -→ F (X ⊗ Y ).

On the other hand, we can describe oplax monoidal functors in terms of the cartesian fibrations, namely as commutative triangles

C ⊗ D ⊗ Fin op * F
where the horizontal functor F preserves cartesian morphisms over inert maps in Fin op * . Here the cartesian arrow X ⊗ Y → (X, Y ) in C ⊗ induces the oplax structure map F (X ⊗ Y ) → F (X) ⊗ F (Y ) in D. Moreover, the opposite functor (C ⊗ ) op → Fin * is actually the cocartesian fibration for the natural symmetric monoidal structure on C op , so an oplax monoidal functor can also be described as a lax monoidal functor C op → D op , as the name suggests.

The equivalence of Proposition A A must therefore relate certain fibrewise right adjoint functors between cocartesian fibrations to certain fibrewise left adjoint functors between the corresponding cartesian fibrations. This turns out to be the correct picture for families of adjunctions in general -more precisely, the first main result of this paper is the following: Theorem B. Let B be an ∞-category. Then there is a canonical equivalence of (∞, 2)-categories

Cocart lax,R (B)
Cart opl,L (B op )

(1,2)-op extracting adjoints fibrewise; here the left-hand side denotes the (∞, 2)-category with cocartesian fibrations over B as objects, fibrewise right adjoint functors (that need not preserve cocartesian lifts) as 1-morphisms, and natural transformations between these as 2-morphisms. The right hand side is defined dually, using cartesian fibrations and fibrewise right adjoints, with the directions of 1-and 2-morphisms reversed by the superscript.

Taking B to be an ∞-operad, this specialises to give:

Corollary C. For any ∞-operad O, the extraction of adjoints gives a canonical equivalence of (∞, 2)-categories

MonCat lax,R O MonCat opl,L O (1,2)-op
, where the left-hand side denotes the (∞, 2)-category of O-monoidal ∞-categories, lax O-monoidal functors that admit (objectwise) left adjoints, and O-monoidal transformations; the right-hand side is defined dually using oplax O-monoidal functors that admit right adjoints.

Proposition A A is contained in this statement by taking O = E ∞ and examining the morphism ∞-categories between two symmetric monoidal ∞-categories C and D. We also use Corollary C C to extend a result of Hinich: We show that the internal mapping functor In order to prove Theorem B B, we need to describe functors to the full subcategories Cocart lax (B) and Cart opl (B) of cocartesian and cartesian fibration in Cat/B in terms of fibrations.

We show that functors A → Cocart lax (B) correspond under covariant unstraightenening to functors p = (p 1 , p 2 ) : E → A × B such that (1) p 1 is a cocartesian fibration,

(2) p 1 -cocartesian morphisms map to equivalences under p 2 , (3) for every a ∈ A the functor (p 2 ) a : E a → B on fibres over a is a cocartesian fibration.

We call such a functor a Gray fibration, for reasons that will become clear in a moment. Dually, functors A op → Cocart lax (B) correspond under contravariant unstraightening to functors p = (p 1 , p 2 ) : E → A × B such that (1) p 1 is a cartesian fibration,

(2) p 1 -cartesian morphisms map to equivalences under p 2 ,

(3) for every a ∈ A the functor (p 2 ) a : E a → B on fibres over a is a cocartesian fibration.

We call these functors curved orthofibrations. While the notion of Gray fibrations admits a cartesian dual, op-Gray fibrations, which encode functors A op → Cart opl (B), the key point for our proofs is that curved orthofibrations are self-dual in the sense that they can also be characterised by p 2 being a cartesian fibration, p 2 -cartesian morphisms mapping to equivalences under p 1 and the functors (p 1 ) b : E b → A being cocartesian fibrations. They can therefore also be straightened covariantly in the second variable, and hence correspond to functors B → Cart opl (A).

Combining these one-variable straightenings, we obtain the following "dualisation" equivalences:

Theorem D. There is a natural equivalence of ∞-categories Gray(A, B) CrvOrtho(A op , B) and OpGray(A, B) CrvOrtho(A, B op );

here Gray(A, B) and OpGray(A, B) are the ∞-category of Gray fibrations and op-Gray fibrations over A × B, respectively, while CrvOrtho(A op , B) is the ∞-category of curved orthofibrations over A op × B and in both cases the morphisms are required to preserve the defining (co)cartesian morphisms.

Special cases of this duality were already known. For example, bifibrations are precisely those curved orthofibration whose fibres are ∞-groupoids, and under the equivalences above they correspond precisely to the left and right fibrations, respectively. An equivalence of this kind was first established by Stevenson in [START_REF] Stevenson | Model structures for Correspondences and Bifibrations[END_REF]] by different means.

To see the relation of these results on two-variable fibrations to Theorem B B, let us explain how the equivalence we build acts on a morphism f : D → E of cartesian fibrations over B which is given fibrewise by left adjoints:

(1) First, f can be covariantly unstraightened to a curved orthofibration over B × [1].

(2) Using Theorem D D, this corresponds to a Gray fibration over B op × [1]. Furthermore, because f is given fibrewise by left adjoints, this Gray fibration is also a curved orthofibration over [1] × B op .

(3) Therefore, this new curved orthofibration can be covariantly straightened to a functor [1] op → Cocart lax (B op ), corresponding to a morphism E ∨ → D ∨ over B op between the cocartesian fibrations dual to the cartesian fibrations we started with.

Here the second half of the second step is the parametrised analogue of the statement that adjunctions among ∞-categories can be encoded by functors to [1] that are both cocartesian and cartesian fibrations, with the left and right adjoint obtained by cocartesian and cartesian straightening, respectively.

Our second main result relates (op)lax natural transformations between functors A → Cat to morphisms between the corresponding (co)cartesian fibrations. To see why these should be related, where p and q are cocartesian fibrations, but f is not required to preserve cocartesian morphisms. Given a morphism β : b → b in B and x ∈ E b , the image under f of the cocartesian morphism x → β ! x factors uniquely through the cocartesian morphism f (x) → β ! f (x), giving a commutative triangle

β ! f (x) f (x) f (β ! x)
The resulting morphism β ! f (x) → f (β ! x) is natural in x ∈ E b , and so gives a natural transformation in the square

E b E b E b E b . f b β ! β ! f b
A lax natural transformation between the straightenings of p and q should precisely be given by such squares (together with coherence data for compositions). Note that these squares commute precisely when f preserves cocartesian morphisms, in which case we indeed expect to get an ordinary natural transformation.

If p and q are instead cartesian fibrations, we similarly obtain for φ : b → b an oplax square

E b E b E b E b , f b φ * φ * f b
which suggests that a morphism between cartesian fibrations should give an oplax natural transformation.

In general, lax and oplax natural transformation between functors of (∞, 2)-categories X → Y can be defined as functors [1] X → Y and X [1] → Y, respectively, where denotes the Gray tensor product. To make the preceding discussion precise we thus need to produce an equivalence between functors [1] → Cocart lax (B) and [1] B → Cat, and similarly in the oplax case. By design the former are captured by Gray fibrations over [1] × B, and using Lurie's locally cocartesian straightening equivalence from [START_REF]∞, 2)-categories and the Goodwillie calculus I[END_REF]] and the model of the Gray tensor product introduced by Gagna, Harpaz and Lanari in [GHL21 GHL21], we show: Theorem E. There are natural equivalences of ∞-categories Gray(A, B) Fun(A B, Cat), and consequently natural equivalences of (∞, 2)-categories Cocart lax (B) Fun lax (B, Cat), Cart opl (B) Fun opl (B op , Cat),

given on objects by straightening of (co)cartesian fibrations; here the targets are defined as adjoints to the Gray tensor product in the two variables, and so have functors as objects, (op)lax natural transformations as morphisms, and modifications between these as 2-morphisms.

Using Theorem E E we obtain the following reformulation of Theorem B B:

Corollary F. Extracting adjoints gives a natural equivalence of (∞, 2)-categories

Fun lax,R (B, Cat) Fun opl,L (B, Cat) (1,2)-op
for every ∞-category B, where the superscript R denotes the locally full (or 1-full) sub-2-category of Fun lax (B, Cat) spanned by those lax natural transformations that admit pointwise left adjoints, and dually for the right hand side.

In this formulation, it is clear that the equivalence of Theorem B B should encode a highercategorical form of the calculus of mates, in the following sense: If we have a lax square

A B A B g α β g
of ∞-categories where g and g have left adjoints f and f , then we can define a mate (or Beck-Chevalley) transformation f β → αf as the composite

f β -→ f βgf -→ f g αf -→ αf,
using the unit for the adjunction f g and the counit for f g . This gives an oplax mate square

B A B A . f β α f
Conversely, given such an oplax square where f and f have right adjoints g and g , one can obtain the lax square above as a mate using the counit for the adjunction f g and the unit for f g . It is easy to see that these constructions are inverse to each other, giving an equivalence between lax squares where the horizontal morphisms are right adjoints and oplax squares where they are left adjoints.

We will show that the equivalence of (∞, 2)-categories in Corollary F F is indeed given on 1-morphisms by a functorial version of this "mate correspondence": it takes a lax natural transformation θ R : [1] B → Cat such that θ R b is a right adjoint for every b ∈ B to an oplax transformation θ L : B [1] → Cat such that the lax naturality squares for θ R are the mates of the oplax naturality squares for θ L . Furthermore, we elaborate on the equivalences of Theorem B B and Corollary F F by describing the (co)unit of a fibrewise adjunction fibrationally, and consequently also the passage to adjoint morphisms in families, and we also provide a characterisation of fibrewise adjoints in terms of mapping functors, to identify these in practice.

Let us finally note that the procedure of taking mates should make sense in any (∞, 2)-category, which suggests that a version of Corollary F F should hold with B and Cat replaced by arbitrary (∞, 2)-categories. This generality is not within reach of our methods, but two possible approaches to the general statement are sketched in [ [START_REF] Gaitsgory | A study in derived algebraic geometry I. Correspondences and duality[END_REF].

Remark. This article is one part of a recombination of our earlier preprints [START_REF]A fibrational mate correspondence for ∞-categories[END_REF]] and [HLN20 HLN20], which contain many of the results we present here, most of them twice; the other part will appear as [ [START_REF] Haugseng | Two-variable fibrations and ∞-categories of spans[END_REF]]. For the reader interested in archaeology we mention that Theorems 1.1, 1.2 and 1.3 from [START_REF]A fibrational mate correspondence for ∞-categories[END_REF]] are now contained in Corollary F F, Proposition A A and Proposition 3.4.9 3.4.9, whereas Theorems A, B and C from [HLN20 HLN20] are now part of Proposition A A, Theorem D D and Corollary C C, respectively.

Organisation. Section 2 2 introduces curved orthofibrations and Gray fibrations in more detail and establishes their basic properties. In particular, we deduce Theorem D D as Theorem 2.5.1 2.5.1. In Section 3 3 we then introduce and study parametrised adjunctions in fibrational form. We prove Theorem B B as Theorem 3.1.11 3.1.11, and deduce Proposition A A and Corollary C C as Corollary 3.4.8 3.4.8 and Theorem 3.4.7 3.4.7, respectively. In between this section also contains the identification of the functor in Theorem B B on morphisms with the Beck-Chevalley construction and the characterisation of parametrised adjoints in terms of mapping ∞-groupoids. Section 4 4 then discusses units and counits for parametrised adjunctions and derives the functoriality of the passage to adjoint morphisms in the parametrised context. Finally, in Section 5 5 we establish the connection to lax natural transformations, prove Theorem E E as a combination of Corollary 5.2.9 5.2.9 and Theorem 5.3.1 5.3.1, and lastly deduce Corollary F F as Theorem 5.3.5 5.3.5.

Conventions. As mentioned above, in order to declutter notation we will write Gpd, Cat and Cat 2 for the ∞-categories of ∞-groupoids (or spaces), ∞-categories and (∞, 2)-categories, respectively. By default we use complete two-fold Segal spaces as the definition of the latter, but we will also need to discuss other models in Section 5 5.

The letter ι will denote the core of an ∞-category, i.e. the ∞-groupoid spanned by its equivalences. By a subcategory of an ∞-category we mean a functor such that the induced morphisms on mapping ∞-groupoids and cores are inclusions of path components. A subcategory is full if the functor furthermore induces equivalences on mapping ∞-groupoids, while it is wide if the functor induces an equivalence on cores. Similarly, a sub-2-category of an (∞, 2)-category is a functor inducing subcategory inclusions on mapping ∞-categories and a monomorphism on underlying ∞-groupoids; we say such a sub-2-category is 1-full if it locally full, i.e. is given by full subcategory inclusions on mapping ∞-categories.

Throughout, we shall use small caps such as Cat to indicate the large variants of ∞-categories and boldface such as Cat to indicate the (∞, 2)-categorical variants. We have also reserved suband superscripts on category names to refer to changes on morphisms, e.g. Cart(A) ⊆ Cart opl (A).

We will write Ar(C) for the arrow ∞-category Fun([1], C) of an ∞-category C, and Tw (C) and Tw r (C) for the two versions of the twisted arrow category, geared so that the combined source-target map defines a left fibraton in the former, and a right fibration in the latter case, see 2.5.8 2.5.8.

Acknowledgments

Two-variable fibrations

Our main goal in the present section is to introduce two new classes of fibrations over a product of two ∞-categories, namely curved orthofibrations and Gray fibrations, and describe how they can be unstraightened over one of the two factors, and consequently dualised. We first recall some basic material about (co)cartesian fibrations in §2.1 2.1. Then in §2.2 2.2 we discuss functors to a product of two ∞-categories that behave like a (co)cartesian fibration in one of the two variables; both curved orthofibrations and Gray fibrations are special cases of such functors. In §2.3 2.3 we then introduce (curved) orthofibrations and study their partial unstraightenings. We consider Gray fibrations in §2.4 2.4 and characterise cocartesian and left fibrations among these. Finally, in §2.5 2.5 we record the various ways in which these fibrations can be dualised. In particular, we prove Theorem D D there.

2.1. Background. For the reader's convenience, we begin by briefly reviewing some basic material on (co)cartesian morphisms and fibrations.

2.1.1. Definition. Let p : X → S be a functor of ∞-categories. Then a morphism α :

y → z of X is p-cartesian if the square Map X (x, y) Map X (x, z) Map S (p(x), p(y)) Map S (p(x), p(z)) α * p(α) *
is a pullback square in Gpd for every x ∈ X. Dually, the morphism α is p-cocartesian if it is p op -cartesian when regarded as a morphism in X op , or in other words if for every x ∈ X the square Map X (z, x) Map X (y, x)

Map S (p(z), p(x)) Map S (p(y), p(x)) α * p(α) * is cartesian.
2.1.2. Notation. To make diagrams more readable, for p : X → B we will sometimes indicate a p-cocartesian morphism of X by x y and a p-cartesian morphism of X by x y.

2.1.3. Definition. Let p : X → S be a functor of ∞-categories. If T is a subcategory of S, we say that X has all p-cartesian lifts over T if for every morphism f : a → b in T and every object x such that p(x) b, there exists a filler in the commutative square

[0] X [1] S x d0 p f
which is a p-cartesian morphism. Dually, X has all p-cocartesian lifts over T if X op has all p op -cartesian lifts over T op . The functor p : X → S is a cartesian fibration if X has all p-cartesian lifts over S, and a cocartesian fibration if X has all p-cocartesian lifts over S.

2.1.4. Notation. We will write Cocart lax (S) and Cart opl (S) for the full subcategories of Cat/S spanned by the cocartesian and cartesian fibrations, respectively, and Cocart(S) and Cart(S) for the wide subcategories thereof in which morphisms are required to preserve (co)cartesian edges.

2.1.5. Remark. The definition above is an invariant version of the definition for quasicategories given by Lurie in [Lu09a Lu09a, Definition 2.4.2.1]. More precisely, a map p between quasicategories corresponds to a (co)cartesian fibration in our sense if and only if for some (and then any) factorisation of p into a categorical equivalence followed by a categorical fibration the latter is a (co)cartesian fibration in Lurie's sense.

2.1.6. Definition. Let p : X → S be a functor of ∞-categories.

A morphism α : y → z in X is locally p-(co)cartesian if it is a (co)cartesian morphism for the pullback X × S [1] → [1] of p along p(α) : [1] → S. The functor p is a locally (co)cartesian fibration if the pullback X × S [1] → [1] is a (co)cartesian fibration for every map [1] → S.
2.1.7. Notation. We write LocCocart lax (S) and LocCart opl (S) for the full subcategories of Cat/S spanned by the locally cocartesian and locally cartesian fibrations, respectively. We also denote by LocCocart(S) and LocCart(S) the wide subcategories of these where morphisms are required to preserve locally (co)cartesian morphisms.

2.1.8. Definition. We call a functor p : X → S that is both a cartesian and a cocartesian fibration a bicartesian fibration. We write Bicart (op)lax (S) for the full subcategory of Cat/S spanned by the bicartesian fibrations.

2.1.9. Remark. In the category theory literature our "bicartesian fibrations" are often called "bifibrations"; we will instead use the latter term as in [Lu09a Lu09a], see Definition 2.3.14 2.3.14.

We recall the following characterisation from [Lu09a Lu09a, Lemma 2.4.2.7] of cartesian morphisms in a locally cartesian fibration, which will be used repeatedly below.

2.1.10. Proposition. Suppose p : E → B is a locally cartesian fibration. Then the following are equivalent for a locally p-cartesian morphism f : x → y in E:

(1) f is a p-cartesian morphism.

(2) For every locally p-cartesian morphism g : z → x, the composite f g : z → y is also locally p-cartesian. The following is [Lu09a Lu09a, Proposition 2.4.2.4]:

2.1.12. Lemma. The following conditions on a cartesian fibration p : X → S are equivalent:

(1) the fibres X s are ∞-groupoids for all s in S,

(2) all morphisms in X are p-cartesian,

(3) p is conservative.

→ Y r → Z, a morphism in X whose image in Y is r-cocartesian is q-cocartesian if and only if it is rq-cocartesian.
2.2.2. Corollary. The following are equivalent for a functor p = (p 1 , p 2 ) : X → A × B:

(1) X has all p-cocartesian lifts over A × ιB.

(2) p 1 is a cocartesian fibration and all p 1 -cartesian morphisms lie over equivalences in B.

(3) In the commutative triangle

X A × B A p p1 pr 1
the map p 1 is a cocartesian fibration, and p takes p 1 -cocartesian morphisms to pr 1 -cocartesian morphisms.

Proof. The equivalence of (1) and ( 2 For later use we also note the following consequence of Proposition 2.2.1 2.2.1 here: 2.2.7. Corollary. Let I be an ∞-groupoid and C an ∞-category. The following are equivalent for a functor p : X → I × C:

(1) p is a cocartesian fibration.

(2) For every i ∈ I, the morphism on fibres p i : X i → C is a cocartesian fibration.

(3) The composite

X p → I × C → C is a cocartesian fibration.
Proof. The equivalence of (1) and (3) follows from Proposition 2.2.1 2.2.1, while (1) implies (2) since cocartesian fibrations are closed under base change. Finally, it is easy to check directly that any p icocartesian lift of some morphism in C is also (pr 2 • p)-cocartesian, so (1) implies (3). Alternatively, one can for example apply the criterion of [HMS20 (3) every p 1 -cartesian morphism in X lies over an equivalence in B, (4) every p 2 -cocartesian morphism in X lies over an equivalence in A.

2.3.3. Proposition. The following are equivalent for a functor p = (p 1 , p 2 ) : X → A × B:

(1) p is a curved orthofibration.

(2) In the commutative triangle

X A × B A, p p1 pr 1
p 1 is a cartesian fibration, p takes p 1 -cartesian morphisms to pr 1 -cartesian morphisms, and for every a ∈ A the map on fibres X a → B is a cocartesian fibration.

(3) p is cartesian over A and p r : X r → ιA × B is a cocartesian fibration.

(4) In the commutative triangle

X A × B B, p p2 pr 2
p 2 is a cocartesian fibration, p takes p 2 -cocartesian morphisms to pr 2 -cocartesian morphisms, and for every b ∈ B the map on fibres X b → A is a cartesian fibration.

(5) p is cocartesian over B and p : X → A × ιB is a cartesian fibration.

Proof. The equivalence of (2) (2) and (3) (3), as well as of (4) (4) and (5) (5), follows from Remark 2.2.9 2.2.9 and Corollary 2.2.2 2.2.2. It thus remains to show that (1) (1) is equivalent to one of these pairs, since they correspond to each other under taking opposites.

If p is a curved orthofibration then it is immediate from the definition that p is a cartesian and p r a cocartesian fibration, i.e. (1) (1) implies (3) (3) and (5) (5). Conversely, the implication (2) (2) ⇒ (1) (1) is [HMS20 HMS20, Lemma A.1.10]. For completeness, we also include a brief argument that (5) (5) implies (1) (1): We need to show that a p -cartesian lift λ : x → y in X of an arrow (α, β) : (a, b) → (a , b ), for which β : b → b is an equivalence, is automatically p-cartesian.

Consider thus the black part of the diagram

z (c, d) w x y (c, b) (a, b) (a , b ), λ (α,β)
which is a lifting problem in which one has to find a black dashed arrow in an essentially unique manner. First take an (essentially unique) p-cocartesian lift z -→ w of (c, d) → (c, b). Since this arrow is cocartesian in all of X, there is an essentially unique dotted red arrow lifting the outer triangle on the right. Since the lower horizontal part of the diagram lives over A × ιB there now exists an essentially unique map w -→ x (not drawn) lifting the lower triangle. The composition with z -→ x is the desired black dotted map, and using that z -→ w is p-cocartesian one can then complete the diagram in an essentially unique way. The essential uniqueness of the map z -→ x is seen by reading the argument in reverse. Proof. Item (2) (2) and Item (4) (4) of Proposition 2.3.3 2.3.3 immediately imply the result at the level of objects. We will exhibit the claim on morphisms for the left equivalence, the right being dual. Suppose that

X Y A × B p f q
is the unstraightening of a map η ∈ Fun(A op , Cocart lax (B)). Because it is given by cartesian unstraightening, it will necessarily preserve cartesian edges over A. Naturality of unstraightening implies that the components of η preserving cocartesian edges is equivalent to the map f a on fibres preserving cocartesian edges. Proposition 2.2.1 2.2.1 then implies that f r is a map of cocartesian fibrations. However, we have seen in Proposition 2.3.3 2.3.3 that for a curved orthofibration, the p r -cocartesian edges agree with the p-cocartesian edges over B. Therefore, η preserves cocartesian edges pointwise if and only if f preserves cocartesian edges over B, and is thus a map of curved orthofibrations.

We saw above that curved orthofibrations over A × B can be straightened to functors A op → Cocart lax (B) or B → Cart opl (A). Our next goal is to introduce a further condition that will ensure these functors actually land in the subcategories Cocart(B) and Cart(A) (and thus encode functors A op × B → Cat), giving the notion of orthofibrations. We also specialise further to bifibrations as considered by Lurie in [Lu09a Lu09a, Section 2.4.7], and studied in detail for example in Stevenson [St18 St18] or [HMS20 HMS20, Appendix A]; these straighten to functors A op → LFib(B) and B → RFib(A). [2]

[1].

∂1 σ1

Note that this diagram is characterised by a universal property in Ar(

Cocart lax ([1])) cc : for each object g : X -→ Y in Ar(Cocart lax ([1])) cc , evaluation at {0} ∈ [1] = ρ(0) yields a natural equivalence to the fibre of X over 0 Map Ar(Cocart lax ([1])) cc (ρ, g) ιX 0 .
Indeed, unravelling the definitions shows that a natural transformation ρ ⇒ g whose components preserve cocartesian arrows is given by a cocartesian arrow β : x -→ β ! x in X over β, together with a factorisation of g( β) into a cocartesian morphism followed by a fibrewise one,

(2.3.7) g( β) : g(x) β ! g(x) g(β ! x). ρ β (x)
Now the cartesian unstraightening of ρ over A = [1] is the curved orthofibration

q : Q -→ [1] op × [1]
where Q is the poset given by 11 10 11 00 01 and the projection is the evident one, sending 11 -→ 11 to the identity. Then

Q -→ [1] op × [1]
has the following universal property: for every curved orthofibration p : X -→ A × B and every α : a -→ a and β : b -→ b , there is a natural equivalence between the ∞-groupoid ι(X (a,b) ) of objects in the fibre over (a, b) and the ∞-groupoid of maps of curved orthofibrations

Q X [1] op × [1] A × B.
(id, β) ! (α, id) * x (α, id) * x (α, id) * (id, β) ! x x (id, β) ! x
where we choose p-(co)cartesian morphism and (dotted) compositions as indicated, and finally the dashed arrow is given by either factoring the horizontal dotted morphisms through the cocartesian morphism over (id, β), or equivalently by factoring the vertical dotted morphism through the cartesian morphism over (α, id).

2.3.8. Definition. Let p : X -→ A × B be a curved orthofibration. We will refer to a morphism in X as p-interpolating if it arises as the evaluation at 11 → 11 of a p-interpolating diagram Q → X.

2.3.9. Remark. The p-interpolating edges in X are precisely the edges that arise under unstraightening from the morphisms ρ β (x) described in (2.3.7 2.3.7).

2.3.10. Definition. A functor p = (p 1 , p 2 ) : X → A × B is an orthofibration if it is a curved orthofibration and all p-interpolating morphisms in X are invertible, i.e. for every pair of morphisms α : a → a in A and β : b → b in B and every object x in X over (a, b), the interpolating morphism

(id, β) ! (α, id) * x -→ (α, id) * (id, β) ! x
is an equivalence. We write Ortho(A, B) for the full subcategory of CrvOrtho(A, B) spanned by the orthofibrations.

2.3.11. Proposition. The following are equivalent for a curved orthofibration p = (p 1 , p 2 ) : X → A × B:

(1) p is an orthofibration.

(2) For every morphism α : a → a in A the cartesian transport functor α * : X a → X a preserves p 2 -cocartesian morphisms.

(3) For every morphism

β : b → b in B the cocartesian transport functor β ! : X b → X b preserves p 1 -cartesian morphisms.
Proof. Let us consider a curved orthofibration p, morphisms α : a -→ a and β : b -→ b , and x ∈ X (a,b) . By Construction 2.3.5 2.3.5, the associated interpolating morphism fits into commuting triangles

(α, id) * x (id, β ! )(α, id) * x (id, β ! )(α, id) * x (α, id) * (id, β ! )x (α, id) * (id, β ! )x (id, β) ! x. α * ( β) β ! ( α)
Here the diagonal morphisms are the image of the cocartesian morphism β : x -→ (id, β) ! x under α * : X a -→ X a and the image of the cartesian morphism (α, id

) * : (id, α) * x -→ x under β ! : X b -→ X b .
It follows that the interpolating morphism is an equivalence if and only if these images remain cocartesian and cartesian, respectively. This shows that condition (1) (1) is equivalent to both (2) (2) and (3) (3).

From this we see that restricting the equivalence of Corollary 2.3.4 2.3.4 to orthofibrations gives: 2.3.12. Corollary. Straightening over A and B gives natural equivalences

Fun(A op , Cocart(B)) Ortho(A, B) Fun(B, Cart(A)).
Of course one can now apply the another instance of the straightening functor on both outer terms. We will discuss the result in §2.5 2.5 below. For now, let us instead specialise the discussion further. Since interpolating edges always lie over equivalences in A × B, we find: 2.3.13. Proposition. For a functor p : X → A × B the following are equivalent:

(1) p is a conservative curved orthofibration.

(2) p is a curved orthofibration whose fibres are ∞-groupoids.

(3) p is a curved orthofibration and p is a left fibration.

(4) p is a curved orthofibration and p r is a right fibration.

(5) p 1 is a cartesian fibration and a morphism in X is p 1 -cartesian if and only if it is sent to an equivalence by p 2 , and p 2 is a cocartesian fibration and a morphism in X is p 2 -cocartesian if and only if it is sent to an equivalence by p 1 .

If these conditions are satisfied, then p is in particular an orthofibration.

Definition.

A bifibration is a functor p = (p 1 , p 2 ) : X → A × B satisfying the equivalent conditions of the previous proposition.

Restricting the equivalence of Corollary 2.3.12 2.3.12 to bifibrations gives: 2.3.15. Corollary. Straightening over A and B gives natural equivalences

Fun(A op , LFib(B)) Bifib(A, B) Fun(B, RFib(A)).
2.4. Gray fibrations. We saw above that curved orthofibrations over A op ×B could be unstraightened to functors A → Cocart lax (B). We can also consider the functors to A × B that correspond to such functors under cocartesian unstraightening over A, which leads to the following definition:

2.4.1. Definition. A Gray fibration over (A, B) is a functor p : X → A×B such that p is cocartesian over A and p r : X r → ιA × B is a cocartesian fibration. We write Gray(A, B) for the subcategory of Cat/(A × B) whose objects are the Gray fibrations, with morphisms required to preserve both types of cocartesian morphisms. Dually, we say p : X → A × B is an op-Gray fibration if p op is a Gray fibration over (A op , B op ), and denote the ∞-category they span by OpGray(A, B).

We will see in Corollary 5.2.9 5.2.9 below that Gray fibrations over (A, B) encode functors of (∞, 2)categories A B → Cat, where denotes the Gray tensor product, which is the reason for the name. In particular, just as the Gray tensor product is not symmetric, let us point out that a Gray fibration (p 1 , p 2 ) : X -→ A×B typically does not determine a Gray fibration (p 2 , p 1 ) : X -→ B ×A.

2.4.2. Observation. From Corollary 2.2.2 2.2.2 and Remark 2.2.9 2.2.9 we see that a functor p = (p 1 , p 2 ) : X → A × B is a Gray fibration if and only if in the commutative triangle

X A × B A, p p1 pr 1
p 1 is a cocartesian fibration, p takes p 1 -cocartesian morphisms to pr 1 -cocartesian morphisms, and for every a ∈ A the map on fibres X a → B is a cocartesian fibration.

Combining this observation with Corollary 2.2.4 2.2.4, and the same analysis as in 2.3.4 2.3.4, we see:

2.4.3. Corollary. Straightening over A gives a natural equivalence

Gray(A, B) Fun(A, Cocart lax (B)) cc .
Our next goal is to give an alternative characterisation of Gray fibrations, namely as those locally cocartesian fibrations that are cocartesian over certain triangles in the base. This characterisation will be the key to relating them to Gray tensor products below in §5.2 5.2. We first observe that Gray fibrations are in particular locally cocartesian fibrations:

2.4.4. Lemma. Let p : X → A × B be a Gray fibration. Then every morphism in X over (α, β) of the form x → (id, β) ! x → (α, id) ! (id, β) ! x,
where the first morphism is p-cocartesian over (id, β) and the second is p r -cocartesian over (α, id), is locally p-cocartesian. In particular, p is a locally cocartesian fibration where all locally p-cocartesian morphisms are of this form, and we have a fully faithful inclusion

Gray(A, B) ⊆ LocCocart(A × B).
Proof. It follows from Proposition 2.1.10 2.1.10 that the morphisms of the given form are locally pcocartesian, since any p r -cocartesian morphism in X r is in particular locally p-cocartesian. Thus X has all locally p-cocartesian lifts, i.e. p is a locally cocartesian fibration. Moreover, all locally p-cocartesian morphisms are of the given form by uniqueness. It remains to show that a morphism f : X → Y between Gray fibrations over A × B preserves locally cocartesian morphisms if and only if it lies in Gray(A, B), which is immediate from the description of locally p-cocartesian morphisms in terms of the two types of cocartesian morphisms for a Gray fibration.

Remark. It is immediate from the definition that any cocartesian fibration over A × B is a Gray fibration. Since Cocart(A × B) is also a full subcategory of LocCocart(A × B) it follows that we have a fully faithful inclusion

Cocart(A × B) ⊆ Gray(A, B).
The following characterisation pins down exactly how Gray fibrations fit in between cocartesian and locally cocartesian fibrations: 2.4.6. Lemma. A locally cocartesian fibration p : X -→ A × B is a Gray fibration if and only if it restricts to a cocartesian fibration over each triangle σ : [2] -→ A × B of one of the following forms:

(a, b) (a, b ) (a, b) (a , b) (a, b) (a , b) (a, b ) (a , b) (a , b ) (id,β) (id,β β) (id,β ) (α,id) (α α,id) (α ,id) (α,id) (α,β) (id,β)
Proof. For a locally cocartesian fibration p, being cocartesian over the first type of triangle is equivalent to p r being a cocartesian fibration by Corollary 2.1.11 2.1.11. Using Proposition 2.1.10 2.1.10, being cocartesian over the second and third types of triangles is equivalent to p being a locally cocartesian fibration such that for any two locally cocartesian arrows x -→ x and x -→ x covering (id, β) and (α, β ) respectively, their composition is locally cocartesian as well. By Proposition 2.1.10 2.1.10, this means precisely that p admits cocartesian lifts over A × ιB.

We see that the difference between Gray and cocartesian fibrations lies in the fact that in a Gray fibration the locally cocartesian lifts of the three edges in a diagram of the form

(a, b) (a , b) (a , b ) (id,β) (α,β) (α,id)
need not form a commutative diagram. We now analyze the relationship between Gray fibrations and cocartesian fibrations over a product more closely.

2.4.7. Construction. Let p : X → A × B be a Gray fibration. Consider an edge (α, β) : (a, b) → (a , b ) in A×B as above. Given a lift x of the source of this edge we can choose p r -and p-cocartesian lifts as in the solid part of Note that we do not include in the notation whether an edge in X is regarded as an interpolating edge for a Gray or curved orthofibration, assuming p is both (a situation we will have to explicitly consider later). We will be more explicit when the need arises.

(id, β) ! (α, id) ! x (α, id) ! x (α, id) ! (id, β) ! x x (id, β) ! x.
2.4.9. Proposition. Let p : X → A × B be a Gray fibration. Then the following are equivalent:

(1) p is a cocartesian fibration,

(2) p restricts to a cocartesian fibration over each triangle σ :

[2] → A × B of the form (a, b) (a, b ) (a , b ), (id,β) (α,β) (α,id)
(3) Every p-interpolating edge in X is an equivalence.

(4) For every morphism α : a → a in A, the cocartesian transport functor α ! : X a → X a preserves cocartesian morphisms over B.

( the composite x 00 → x 11 → x 22 of locally p-cocartesian morphisms over (α, β) and (α , β ) is again locally p-cocartesian over (α α, β β). We can expand this to a composite

x 00 → x 10 → x 11 → x 12 → x 22
of locally p-cocartesian morphisms over (α, id), (id, β), (id, β ), and (α , id). Then the composite x 10 → x 12 is locally p-cocartesian over (id, β β), and so the composite morphism x 00 → x 12 can alternatively be factored as x 00 → x 02 → x 12 where these morphisms are locally p-cocartesian over (id, β β) and (id, α). Then x 02 → x 12 → x 22 is locally p-cocartesian over (α α, id), and so finally x 00 → x 02 → x 22 is locally cocartesian over (α α, β β) as required. Hence p is a cocartesian fibration by Corollary 2.1.11 2.1.11.

From Construction 2.4.7 2.4.7 we see that (2) implies (3), since the p-interpolating morphisms are now obtained by factoring a morphism that is already locally p-cocartesian. Moreover, if all p-interpolating morphisms are invertible we can also conclude that the composite of locally pcocartesian morphisms over a triangle as in (2) factors as a locally p-cocartesian morphism followed by an equivalence, and hence is again locally p-cocartesian.

Since the cocartesian edges in X a over B are precisely the locally p-cocartesian edges in X that lie over id a , the equivalence of (3) and ( 4) is immediate from the definition of p-interpolating edges, while (5) is just a rephrasing of (4).

The interpolating edges of a Gray fibration p : X → A × B map to ι(A × B) by construction. From the analogous assertion for cocartesian fibrations we therefore immediately obtain: 2.4.10. Corollary. A Gray fibration p : X → A×B is a left fibration if and only if it is conservative, or equivalently if its fibres are ∞-groupoids.

-→ [1] × [1] from Definition 2.4.8 2.4.8 is dual to the curved orthofibration Q -→ [1] op × [1]
from Construction 2.3.5 2.3.5, so it follows that the ∞-groupoid of interpolating diagrams Q -→ X and Q -→ Y are equivalent. Since dualisation identifies the morphism 11 -→ 11 in Q with 11 -→ 11 in Q , dualisation preserves interpolating morphisms as well.

2.5.6. Corollary. The equivalences from Theorem 2.5.1 2.5.1 restrict to equivalences

Cocart(A × B) Ortho(A op , B) and LFib(A × B) Bifib(A op , B)
and dually

Ortho(A, B) Cart(A × B op ) and Bifib(A, B) RFib(A × B op )
Proof. The left hand equivalences follow by replacing the use of Corollary 2.3.4 2.3.4 and Corollary 2.4.3 2.4.3 in Theorem 2.5.1 2.5.1 with Corollary 2.3.12 2.3.12 and straightening for (co)cartesian fibrations. Alternatively, using the previous proposition they follow from characterisation (3) (3) of Proposition 2.4.9 2.4.9. The statement about left and bifibrations follows by inspecting fibres. (2) From Corollary 2.5.6 2.5.6 we obtain a diagram of equivalences Ortho(A, B)

Cocart(A op × B) Cart(A × B op ) ∼ ∼ ∼
where the lower maps are given by dualisation in a single variable (i.e. over A op × B). It is not a priori clear that this diagram commutes, but this will also be a consequence of the results in [ [START_REF] Haugseng | Two-variable fibrations and ∞-categories of spans[END_REF]]. Combined with the usual straightening equivalences for (co)cartesian fibrations, we similarly obtain two a priori different equivalences

Ortho(A, B) Fun(A op × B, Cat),
given by straightening first over A and then over B, or vice versa. Both restrict to equivalences Bifib(A, B) Fun(A op × B, Gpd) and their agreement seems to be new even in this latter case.

Parametrised and monoidal adjunctions

In the present section we will use the results of Section 2 2 to study the operation of taking adjoint functors in families. The statements we prove in §3.1 3.1 boil down to the fact that for any lax natural transformation g : F ⇒ G between two diagrams of ∞-categories, such that each component of g is a right adjoint, the pointwise left adjoints assemble into an oplax natural transformation f : G ⇒ F . For the moment, we shall, however, stay in the fibrational picture and instead consider maps between the associated (co)cartesian fibrations. In particular, we will prove Theorem B B from the introduction. The translation of this statement into the mate correspondence for (op)lax transformations will be delayed till §5.3 5.3. In §3.2 3.2 and §3.3 3.3 we carry out two consistency checks: In the former we show that for each morphism our functorial passage to fibrewise adjoints is given by the Beck-Chevalley construction on morphisms, and in the latter we prove that the fibrewise adjoints we produce in the fibrational picture are characterised by the expected relation on morphism ∞-groupoids from the left to the right adjoint.

In §3.4 3.4 we then finally specialise the discussion to maps of ∞-operads and produce the correspondence between lax O-monoidal structures on a right adjoint functor and oplax O-monoidal structures on its left adjoint. In particular, we will prove Proposition A A and Corollary C C here. As defined the categories Cocart lax,R (B) and Cart opl,L (B) are oblivious to the fact that there are non-invertible transformations between B-parametrised adjoints. As it is often important not to forget these when passing to adjoints (in the specialisation to symmetric monoidal categories in §3.4 3.4 they correspond to symmetric monoidal natural transformations for example), we first enhance the ∞-categories from Definition 3.1.1 3.1.1 to (∞, 2)-categories that encode natural transformations as their 2-morphisms.

For this we will use the description of (∞, 2)-categories as complete 2-fold Segal ∞-groupoids:

3.1.2. Definition. A complete 2-fold Segal ∞-groupoid is a functor X : ∆ op × ∆ op → Gpd such that
(1) the simplicial ∞-groupoids X n,• and X •,m satisfy the Segal condition for all n, m,

(2) the simplicial ∞-groupoid X 0,• is constant,

(3) the Segal ∞-groupoids X •,0 and X 1,• (and hence X n,• for all n) are complete.

Note that by [JFS17 JFS17, Lemma 2.8] these conditions imply that X •,m is also complete for all m.

We use the following general construction to enhance our ∞-categories to (∞, 2)-categories:

3.1.3. Proposition. Suppose F : Cat op → Cat is a limit-preserving functor such that for every ∞-category B the functor F (|B|) → F (B) arising from the canonical map B → |B| induces a monomorphism ιF (|B|) → ιF (B) on underlying ∞-groupoids. If we define F lc (B) ⊆ F (B) to be the full subcategory spanned by the image of F (|B|) under this functor, then F lc is also a limit-preserving functor Cat op → Cat, and the bisimplicial ∞-groupoid

([n], [m]) → Map Cat ([n], F lc ([m]))
is a complete 2-fold Segal space.

Proof. Since the map B → |B| is a natural transformation in B, we see that F lc is a subfunctor of F . Note that the condition that ιF

(|B|) → ιF (B) is a monomorphism implies that ιF (|B|) → ιF lc (B) is an equivalence. For any colimit of ∞-categories B colim i B i we then have a commutative diagram F (|B|) lim i F (|B i |) F lc (B) lim i F lc (B i ) F (B) lim i F (B i ), ∼ ∼
where the top and bottom horizontal maps are equivalences since F preserves limits (and |-| : Cat → Gpd → Cat preserves colimits), and the bottom right vertical map is fully faithful since fully faithful maps are closed under limits. Hence the middle horizontal functor is also fully faithful, by the 2-of-3 property for equivalences applied to mapping ∞-groupoids. In the top square the vertical morphisms are both given by equivalences on underlying ∞-groupoids, since this condition is also closed under limits. By the 2-of-3 property it follows that the middle horizontal functor is also an equivalence on underlying ∞-groupoids, and hence it is an equivalence. Thus F lc preserves limits.

It follows that the functor

([n], [m]) → Map Cat ([n], F lc ([m]))
satisfies the Segal and completeness conditions levelwise in each variable, since these can be expressed as taking certain colimits in Cat to limits.

(3.1.6) [m], [n] Map Cat [m], Cocart lax [n] B × [n]
is a complete 2-fold Segal ∞-groupoid.

The same assertion holds if we instead take Cocart lax,R S (B×S), Cart opl S (B×S) or Cart opl,L S (B×S), which are all defined analogously.

3.1.7. Definition. Let B be a small ∞-category. We define Cocart lax (B) to be the (∞, 2)-category associated to the complete 2-fold Segal ∞-groupoid (3.1.6 3.1.6). Likewise, we define the (∞, 2)-category Cart opl (B) to be the (∞, 2)-category associated to the 2-fold complete Segal ∞-groupoid

(3.1.8) [m], [n] Map Cat [m], Cart opl [n] B × [n]
We define the (∞, 2)-categories Cocart lax,R (B) and Cart opl,L (B) similarly.

In We now come to our main technical result, Theorem B B: 3.1.11. Theorem. Let B be an ∞-category. Then there is a natural equivalence of (∞, 2)-categories (3.1.12) Adj :

Cocart lax,R (B) ∼ -→ Cart opl,L (B op ) (1,2)-op
sending each cocartesian fibration to the cartesian fibration classifying the same functor B -→ Cat.

Here in the target the directions of 1-and 2-morphisms are changed, as indicated.

In particular, for B = * this produces an equivalence

Cat R (Cat L ) (1,2)-op .
For the proof recall first that a functor g : C -→ D between ∞-categories is a right adjoint if the corresponding cartesian fibration p : X -→ [1] is a cocartesian fibration as well, see [Lu09a Lu09a, Section 5.2.2]. Dually, a functor f : D -→ C is a left adjoint if the corresponding cocartesian fibration is a cartesian fibration as well. In other words, one can encode adjunctions by functors p : X -→ [1] that are simultaneously cartesian and cocartesian fibrations; the two adjoint functors can be extracted from this by (co)cartesian straightening.

We now extend this statement by showing that a functor between two B-parametrised categories (in the form of cocartesian fibrations over B) is a right adjoint if and only if the corresponding curved orthofibration over [1] × B is also a Gray fibration, when considered over B × [1], and similarly in the dual situation. More generally: 3.1.13. Lemma. Let p = (p 1 , p 2 ) : X -→ A × B be a functor. Then the following conditions are equivalent:

(1) p is a curved orthofibration and the functor A op -→ Cocart lax (B) classifying p via Corollary 2.3.4 2.3.4 takes values in the wide subcategory Cocart lax,R (B),

(2) p is a curved orthofibration whose restriction p l to A × ι(B) is a cocartesian fibration as well,

(3) p = (p 2 , p 1 ) : X -→ B × A is a Gray fibration and the functor B → Cocart lax (A) classifying B takes values in the wide subcategory Bicart (op)lax (A), (4) p = (p 2 , p 1 ) : X -→ B × A is a Gray fibration whose restriction to ι(B) × A is a cartesian fibration as well.

Dually, a curved orthofibration q = (q 1 , q 2 ) : Y -→ B × A classifies a functor A -→ Cart opl,L (B) via Corollary 2.3.4 2.3.4 if and only if q r is a cartesian fibration as well, or equivalently if and only if (q 2 , q 1 ) : Y -→ A × B is an op-Gray fibration, which is then automatically classified by a functor A op → Bicart (op)lax (B).

Proof. For the equivalence between (1) (1) and ( 2 Let us write M (A, B) for the ∞-groupoid of functors p : X -→ A × B satisfying the equivalent conditions of Lemma 3.1.13 3.1.13, so that there are natural inclusions of path components

ιGray(B, A) M (A, B) ιCrvOrtho(A, B)
where the left inclusion sends p = (p 1 , p 2 ) to (p 2 , p 1 ). Likewise, let us write N (A, B) for the ∞-groupoid of functors q : Y -→ B × A satisfying the equivalent opposite conditions of Lemma 3.1.13 3.1.13, so that there are natural inclusions of path components ιOpGray(A, B) N (A, B) ιCrvOrtho(B, A).

Un ct : Map Cat [m], Cocart lax,R [n] B × [n] M [n] [m] op , B × [n] Un cc : Map Cat [m], Cart opl,L [n] B × [n] N [n] [m], B × [n] .
Proof. Apply Lemma 3.1.13 3.1.13 and use that local constancy along [n] can be checked when [m] = * , in which case the unstraightening functors are equivalent to the identity.

3.1.15. Lemma. The dualisation functor from Theorem 2.5.1 2.5.1 with respect to B × S

D ct : Gray B × S, A ∼ -→ CrvOrtho (B × S) op , A
restricts to an equivalence of ∞-groupoids

D ct : M S (A, B × S) ∼ -→ N S op (A, (B × S) op ).
Proof. When A = * , dualisation over B × S simply sends cocartesian fibrations to their dual cartesian fibrations. This preserves local constancy in S and by naturality in A, one sees that dualisation preserves those objects that restrict to locally constant fibrations over {b} × S × A.

By the addenda of Theorem 2.5.1 2.5.1, for B × S = * the dualisation equivalence restricts to a natural self-equivalence of the ∞-category of cocartesian fibrations over A that is equivalent to the identity. By naturality in B × S, one therefore sees that the dualisation preserves those objects that restrict for each x ∈ B × S to a bicartesian fibration over {x} × A, as required.

Proof of Theorem 3.1.11 3.1.11. Corollary 3.1.14 3.1.14 and Lemma 3.1.15 3.1.15 yield a natural equivalence (3.1.16)

Map Cat A op , Cocart lax,R S (B × S) M S (A, B × S) Map Cat A, Cart opl,L S op (B op × S op ) N S op (A op , B op × S op ) ∼ Un ct D ct

Str cc

Taking A and S to be simplices, one obtains the desired equivalence between 2-fold Segal spaces Cocart lax,R (B) Cart opl,L (B op ) (1,2)-op .

3.1.17. the prototypical example is the tensor-hom adjunction in a (left-)closed monoidal ∞-category. This is a special case of our parametrised adjunctions: It follows from the Yoneda lemma that given F , the functor G is uniquely determined and exists if and only if F (b, -) is a left adjoint for all b ∈ B. We can then view F as a parametrised left adjoint

B × C B × D B.
(pr 1 ,F ) pr 1 pr 1

Since the dual cocartesian fibration to pr 1 : B × C → B is the projection B op × C → B op , Theorem 3.1.11 3.1.11 produces a parametrised right adjoint in the form

B op × C B op × D B op . pr 1 (pr 1 ,G) pr 1
At the moment we only know that G(b, -) is right adjoint to F (b, -) for each b, but we will verify in Corollary 3.3.16 3.3.16 below that G indeed gives the expected natural equivalence on mapping ∞-groupoids. We will apply this fibrational description of two-variable adjunctions to analyse the monoidal properties of the internal mapping functor in Corollary 3.4.10 3.4.10.

3.2. Identifying mates. Our goal in this subsection is to describe the effect on morphisms of the equivalence from Theorem 3.1.11 3.1.11 in terms of mates or Beck-Chevalley transformations, see Proposition 3.2.7 3.2.7 below.

In order to do this, let us first recollect how one can obtain the unit and counit of the adjunction from a bicartesian fibration p : X -→ [1], using the following general construction: 3.2.1. Construction. Let p : X -→ [1] be a cocartesian fibration and I any ∞-category. By [Lu09a Lu09a, Proposition 3.1.2.1], post-composition with p determines a cocartesian fibration p I : Fun(I, X) → Fun(I, [1]), with cocartesian morphisms those natural transformations that are given by p-cocartesian morphisms at each object of I. Given a functor φ : I → X, its cocartesian transport functor φ cc : I × [1] → X is the diagram corresponding to the essentially unique p I -cocartesian morphism with domain φ covering the map pφ ⇒ const 1 in Fun(I, [1]). Alternatively, it is the unique diagram whose restriction to I × {0} is given by φ such that each φ cc (i) :

[1] -→ X is p-cocartesian over p(φ(i) ≤ 1).
Dually, for a cartesian fibration p : X -→ [1] and a functor ψ : I -→ X one can form the cartesian transport functor ψ ct : I × [1] → X of ψ. The functor (i 0,cc ) ct (-, -, 0) factors through the fibre D X 0 , and encodes the unit transformation η : id D ⇒ gf of the adjunction classified by p. The above square shows that for a fixed object x, the unit η y : y -→ gf (y) is obtained by taking a cocartesian arrow y -→ f (y) and factoring it as a fibrewise map followed by a cartesian map. Dually, starting with the cartesian transport of the fibre inclusion C → X and then taking the cocartesian transport gives (i 1,ct

diagram i 0,cc : D × [1] → X yields a functor (i 0,cc ) ct : D × [1] × [1] → X,
) cc : C × [1] × [1] → X whose restriction to C × [1] × {1} encodes the counit transformation : f g ⇒ id C of the adjunction.
To understand the behaviour of a B-parametrised right adjoint, let us start by showing that a map in Cocart lax (B) can roughly be viewed as a lax natural transformation; this picture will be made more precise in Section 5 5.

β ! g β ! ρ β g
We will refer to this as the β-component of g. To see this, note that for each object x ∈ C(b), the image of the cocartesian lift β : x -→ β ! x under g factors uniquely as

g( β) : g(x) β ! g(x) g(β ! x). ρ β (x)
Alternatively, Example 2.3.5 2.3.5 shows that ρ β (x) can also be obtained as the interpolating edge associated to x in the curved orthofibration p :

X -→ [1] op × B classifying g.
To organise these interpolating morphisms ρ β (x) into a natural transformation, one can use a similar maneuver as in Construction 3.

f β ! f β ! gf f gβ ! f β ! f. f β ! η f ρ β f β ! f
We are now ready to describe the effect of the equivalence Adj from Theorem 3.1.11 3.1.11 on morphisms. To this end, let g : C -→ D be a morphism in Cocart lax,R (B) and let f = Adj(g) be the induced morphism in Cart opl,L (B op ) op . Construction 3.2.4 3.2.4 and the dual analysis for maps in Cart opl (B) show that for each β : b -→ b in B, the maps g and f give rise to lax commuting squares of the form

C(b) D(b) C(b) D(b) C(b ) D(b ) C(b ) D(b ). β ! g β ! ρ β (β op ) * (β op ) * f g f λ β
Note that in these diagrams, the vertical change-of-fibre functors are equivalent. The transformation λ β is given by the Beck-Chevalley transformation associated to ρ β , more precisely: 3.2.7. Proposition. Let g : C -→ D be a map in Cocart lax,R (B), and β : b → b a morphism in B, so that the β-component of g is given by (3.2.5 3.2.5). Then regarding β as a morphism in B op , the β-component of f = Adj(g) is given by the Beck-Chevalley transformation associated to ρ β , i.e. λ β is equivalent to the composition 3.5 2.3.5 identify ρ β (x) with the corresponding p-interpolating morphism β ! g(x) -→ gβ ! (x) in X, where p is considered as a curved orthofibration. On the other hand, for y ∈ D(b) X 0,b , (the dual of) Construction 3.2.4 3.2.4, Definition 2.4.8 2.4.8 and Proposition 2.5.5 2.5.5 show that λ β (y) : f β ! y -→ β ! f y is given by the associated p-interpolating morphism, where p is considered as a Gray fibration.

f β ! f β ! gf f gβ ! f β ! f. f β ! η f ρ β f β ! f
To relate λ β and ρ β , take y ∈ D(b) and consider the following diagram in X (3.2.8)

y gf (y) f (y) β ! gf (y) β ! (y) gβ ! f (y) β ! f (y), η ρ β f β ! η
which we build in steps as follows; first we obtain the outer square as the essentially unique one in which the two vertical arrows are p-cocartesian (as always denoted ) and the map y -→ f (y) is locally p-cocartesian. We then factor the two horizontal maps into a fibrewise map, followed by a p-cartesian morphism (denoted ). Finally, we factor the induced map gf (y) -→ gβ ! f (y) into a cocartesian map followed by a fibrewise one. Note that the right rectangle is then precisely the Q-diagram exhibiting ρ β (f (y)) : β ! gf (y) -→ gβ ! f (y) as a p-interpolating edge (Construction 2.3.5 2.3.5).

The resulting morphism y -→ gf (y) in the top row is the unit of the adjoint pair (f, g) (at the fibre over b) and the map β ! η exists since the left vertical map was p-cocartesian. Now note that the maps β ! η and ρ β f are both contained in the fibre X 0,b D(b ), so that choosing locally p-cocartesian lifts over (0, b ) → (1, b ) yields a commuting diagram

β ! (y) β ! gf (y) gβ ! f (y) f β ! (y) f β ! gf (y) f gβ ! f (y) β ! f (y). β ! η • ρ β f • • f β ! η f ρ β f
Here the top lives in X 0,b and the bottom in X 1,b . Pasting this diagram below (3.2.8 3.2.8), the resulting outer diagram determines a Q -diagram in X (Definition 2.4.8 2.4.8) that exhibits the bottom composite f β ! (y) -→ β ! f (y) as the p-interpolating arrow associated to y. In particular, the bottom composite is equivalent to λ β (y).

To obtain the identification as natural transformations, we proceed as in Construction 3. 3.3. Parametrised correspondences. Our goal in this section is to derive a characterisation of parametrised adjoints, which we defined in terms of their associated fibrations in §3.1 3.1, by means of a natural equivalence analogous to the usual equivalence

Map C (f (d), c) Map D (d, g(c))
on mapping ∞-groupoids associated to an adjunction f g.

To motivate the form this will take, let us first observe that we can phrase the preceding condition in terms of left fibrations: f is left adjoint to g if there is an equivalence

(3.3.1) (f op × id D ) * Tw (D) (id C × g) * Tw (C)
of left fibrations over C op × D, since the twisted arrow ∞-categories are the left fibrations for the mapping ∞-groupoid functors, and precomposition corresponds to pullback of left fibrations. We will prove a parametrised analogue of (3.3.1 3.3.1); to state this we first need some notation: 3.3.2. Notation. To simply a number of formulae we use (-) ∨ to denote the cocartesian fibration dual to a cartesian fibration in this subsection. Now suppose p : E → B is a cocartesian fibration, corresponding to a functor F : B → Cat. The natural transformation Tw (F (-)) → F (-) op × F (-) then corresponds to a commutative triangle

Tw B (E) (E op ) ∨ × B E B, since (E op ) ∨ → B
(f op × B id) * Tw B (C) (id × B g) * Tw B (D) of left fibrations over (D op ) ∨ × B C.
Before we embark upon the proof of Theorem 3. where here LFib denotes the full subcategory of Ar(Cat) spanned by the left fibrations.

We use the following result from [START_REF] Stevenson | Model structures for Correspondences and Bifibrations[END_REF]], see also [AF20 AF20]:

Theorem (Stevenson).

There is an equivalence

corr : Cat/[1] ∼ -→ Corr,
over Cat × Cat, where the functor Cat/[1] → Cat × Cat is given by taking fibres over 0 and 1. The value of the functor corr on f : E → [1] is defined by the natural pullback square corr(E) Tw (E)

E op 0 × E 1 E op × E,
where E 0 and E 1 are the fibres of f over 0 and 1 respectively.

It is easy to check that if E → [1] is in fact a cocartesian fibration, corresponding to a functor f : E 0 → E 1 , then there is a pullback square corr(E)

Tw (E 1 )

E op 0 × E 1 E op 1 × E 1 , f op ×id
while if it is a cartesian fibration, corresponding to g : E 1 → E 0 , then we have a pullback corr(E) Tw (E 0 )

E op 0 × E 1 E op 0 × E 0 .
id×g Combining these squares we get the equivalence (3.3.1 3.3.1) when E → [1] corresponds to an adjunction. We now want to develop a parametrised version of this story.

3.3.6. Definition. A B-parametrised correspondence is a left fibration X → (E op 0 ) ∨ × B E 1 for cocartesian fibrations E 0 , E 1 → B.
We define the ∞-category Corr(B) thereof by the pullback square

Corr(B) LFib Cocart(B) × Cocart(B) Cat. ev1 (-op ) ∨ × B (-)
Using [Lu09a Lu09a, 2.4.2.11] once again, we find that a B-parametrised correspondence is equivalently given by the data of two cocartesian fibrations E 0 → B , E 1 → B and a commutative triangle

X (E op 0 ) ∨ × B E 1 B f p q
between cocartesian fibrations, such that f preserves cocartesian edges and f a is a left fibration for every b ∈ B. Straightening this data in the base B, we get that Corr(B) is equivalently given by the following pullback:

Corr(B) Fun(B, LFib)
Fun(B, Cat) × Fun(B, Cat) Fun(B, Cat).

ev * 1 (-) op ×(-)
Because Fun(B, -) preserves pullbacks, this implies that Corr(B) is equivalent to Fun(B, Corr).

Corollary.

There is an equivalence

corr B : RCocart([1], B) ∼ -→ Corr(B) over Cocart(B) × Cocart(B), where the functor RCocart([1], B) → Cocart(B) × Cocart(B)
is given by taking fibres over 0 and 1. Given

f : E → [1] × B in RCocart([1], B), its value corr B (E) is defined by the natural pullback square corr B (E) Tw B (E) (E op 0 ) ∨ × B E 1 (E op ) ∨ × B E,
where

(E op ) ∨ ∈ RCocart([1], B)
B (E) Tw B (E 0 ) (E op 0 ) ∨ × B E 1 (E op 0 ) ∨ × B E 0 . id× B g
In order to prove this we first make some fibrational observations:

3.3.9. Proposition. Consider a commutative square of ∞-categories E F X Y, g p q f
where p and q are cocartesian fibrations and g takes p-cocartesian morphisms to q-cocartesian ones. For x ∈ X, let g x : E x → F f x be the restriction of g to the fibres over x. If a morphism φ : e → e in E x is g x -cartesian, then φ is also g-cartesian.

Proof. For e ∈ E over x ∈ X, we have the commutative diagram Map E (e , e ) Map E (e , e)

Map X (x , x)
Map F (ge , ge ) Map F (ge , ge)

Map Y (f x , f x),
where we want to show that the back square is cartesian. It suffices to check we have a cartesian square on the fibres over any ξ ∈ Map X (x , x), but since p and q are cocartesian fibrations and g preserves cocartesian morphisms, we can identify this square as Map Ex (ξ ! e , e ) Map Ex (ξ ! e , e)

Map F f x (f (ξ) ! ge , ge ) Map F f x (f (ξ) ! ge , ge),
which is cartesian by the assumption that φ is g x -cartesian. Proof. To find that a morphism in Tw (E), in which x → x is an equivalence and y → y is p-cartesian, is Tw (p)-cartesian, apply Proposition 3.3.9 3.3.9 to the square Tw (E) Tw (B)

E op B op ,
B (E) Tw (A) × B (E op ) ∨ × B E A op × A × B,
where we use that Tw B (A × B) Tw (A) × B. Applying Corollary 3.3.10 3.3.10 fibrewise and appealing to Proposition 3.3.9 3.3.9 again, we get: given by the cartesian morphisms for Tw (E b ) → Tw (A) described above.

3.3.12. Observation. In particular, for any a ∈ A the projection

Tw B (E) × Tw (A) A a/ → A a/
is a cartesian fibration, and Tw B (E) × Tw (A) A a/ → A a/ × B is a curved orthofibration.

3.3.13. Notation.

Given E → [1] × B in RCocart([1], B), we have pullback squares Tw B (E)| 0 Tw B (E) (E op 0 ) ∨ × B E (E op ) ∨ × B E, Tw B (E)| 1 Tw B (E) (E op ) ∨ × B E 1 (E op ) ∨ × B E.
Proof of Proposition 3.3.8 3.3.8. Applying Observation 3.3.12 3.3.12 to E → [1] × B and 0

∈ [1], we see that Tw B (E)| 0 → [1] 0/ ∼ = [1]
is a cartesian fibration. Moreover, from the description of the cartesian morphisms we see that

Tw B (E)| 0 → (E op 0 ) ∨ × B E
is a morphism between cartesian fibrations to [1] that preserves cartesian morphisms.

Taking fibres for Tw B (E)| 0 → (E op 0 ) ∨ × B E over 0 and 1, we get the following diagram where both squares are cartesian

Tw B (E 0 ) Tw B (E)| 0 corr B (E) (E op 0 ) ∨ × B E 0 (E op 0 ) ∨ × B E (E op 0 ) ∨ × B E 1 .
Taking the cartesian transport over [0] → [1] we therefore get a commutative square

corr B (E) Tw B (E 0 ) (E op 0 ) ∨ × B E 1 (E op 0 ) ∨ × B E 0 . id× B g
It remains to show that this square is cartesian, which we can check on fibres since the vertical maps are both left fibrations. To do this we can first restrict to fibres over b ∈ B, where we have the square

corr(E b ) Tw (E 0,b ) E op 0,b × E 1,b E op 0,b × E 0,b , id×g b
which is cartesian because g b is a right adjoint.

3.3.14. Observation.

If p : E → [1] × B is in RCocart([1], B), then we also have (p op ) ∨ : (E op ) ∨ → [1] op × B in RCocart([1] op , B).
Since there is a natural equivalence Tw (C) Tw (C op ) over the permutation C op × C C × C op , we get for a cocartesian fibration X → B a natural equivalence Tw B (X) Tw B ((X op ) ∨ ) over (X op ) ∨ × B X X × B (X op ) ∨ , and hence also

corr B (E) corr B ((E op ) ∨ ) over (E op 0 ) ∨ × B E 1 E 1 × B (E op 0 ) ∨ .
Combining this with Proposition 3.3.8 3.3.8, we get the following dual version thereof:

3.3.15. Corollary. Suppose p : E → [1] × B is in RCocart([1], B). If (p op ) ∨ : (E op ) ∨ → [1] op × B is a curved orthofibration, corresponding to a functor f : E ∨ 0 → E ∨ 1 over B op , then there is a pullback square corr B (E) Tw B (E 1 ) (E op 0 ) ∨ × B E 1 (E op 1 ) ∨ × B E 1 . f op × B id
Proof of Theorem 3.3.3 3.3.3. Suppose g corresponds to the curved orthofibration p :

E → [1] × B. Then p ∨ : E ∨ → [1] × B op is a curved orthofibration over B op × [1] whose cocartesian unstraightening over [1] gives f . Hence (p op ) ∨ : (E op ) ∨ → [1] op × B
(id × B g) * Tw B (D) corr B (E) (f op × B id) * Tw B (C) ∼ ∼ of left fibrations over (D op ) ∨ × B C.
Specialising this to the case of projections, we get:

(id × G) * Tw (X) (F op × id) * Tw (Y )
of left fibrations over X op × Y × B, and hence a natural equivalence of mapping spaces

Map X (x, G(y, b)) Map Y (F (x, b), y).
3.4. Lax monoidal adjunctions. Recall that an ∞-operad O is a map of ∞-categories p : O -→ Fin * to the 1-category of pointed finite sets, satisfying the following conditions:

(1) O has all p-cocartesian lifts for inert morphisms in Fin * (i.e. those maps which are bijections away from the basepoint).

(2) Let x ∈ O be an object with p(x) = n and let ρ i x : x -→ x i be a p-cocartesian lift of the unique inert map ρ i : n -→ 1 which sends i to 1. For every f : m -→ n and y ∈ O m , postcomposition with the ρ i

x induce an equivalence

Map f O y, x i Map ρ i •f O (y, x i ).

∼

(3) For every tuple (x 1 , . . . , x n ) of objects in O 1 , there exists an x ∈ O n together with pcocartesian lifts ρ i

x : x -→ x i . A morphism in O is called inert if it is the cocartesian lift of an inert map in Fin * .
If O is an ∞-operad, then an O-monoid in an ∞-category X with finite products is a functor M : O → X satisfying the Segal condition: for every x ∈ O n with inert maps ρ i

x : x -→ x i , the functor M (x) → i M (x i ) is an equivalence. An O-monoidal ∞-category is a cocartesian fibration corresponding to an O-monoid in Cat (or equivalently an ∞-operad with a map to O that is a cocartesian fibration). O has objects symmetric monoidal ∞-categories, 1-morphisms lax symmetric monoidal functors, and 2-morphisms symmetric monoidal transformations. In particular, Theorem A from the introduction is a statement about morphism categories therein (and in the oplax analogue defined below). Note in particular that the objects in MonCat opl O are a priori not O-monoidal ∞-categories: one has to take the cocartesian fibration over O dual to a cartesian fibration over O op to get an O-monoidal ∞-category in the usual sense. The following lemma thus simply asserts that essentially by definition, an oplax O-monoidal functor is a lax O-monoidal functor between the opposite categories.

3.4.4. Lemma. Taking opposite categories defines an equivalence of (∞, 2)-categories

(-) op : MonCat opl O -→ MonCat lax O 2-op .
Proof. It suffices to verify that the equivalence of Remark 3.1.10 3.1.10 identifies the relevant sub-2categories. Given a cartesian fibration C ⊗ -→ O op , let us write C ⊗ -→ O for the opposite cocartesian fibration. The Segal map

(ρ * i ) i : C ⊗ (x) -→ i C ⊗ (x i )
is then the opposite of the Segal map (ρ i,! ) i :

C ⊗ (x) -→ i C ⊗ (x i ) xi
, so that one is an equivalence if and only if the other is. Finally, a functor preserving cartesian lifts of inert morphisms is sent to the functor between opposite categories, which preserves cocartesian lifts of inert morphisms.

For example, for O = (Fin * ) inert , the trivial operad, MonCat opl O is a 2-fold Segal space model for the (∞, 2)-category Cat (by the discussion in Section 5.2 5.2). In this case we find, unsurprisingly, that taking opposite categories defines an equivalence Cat Cat 2-op . 3.4.5. Lemma. Let g : C ⊗ -→ D ⊗ be a lax O-monoidal functor, i.e. a morphism in MonCat lax O . Then the following two conditions are equivalent:

(1) For every x ∈ O 1 , the induced map on fibres g :

C ⊗ (x) -→ D ⊗ (x) is a right adjoint.
(2) For every x ∈ O, the induced map on fibres g :

C ⊗ (x) -→ D ⊗ (x) is a right adjoint.
Proof. This follows from the fact that for each x ∈ O, there is a commuting square

C ⊗ (x) D ⊗ (x) i C ⊗ (x i ) i D ⊗ (x i ) gx ∼ (ρ i ! )i ∼ (ρ i ! )i (gx i )
where ρ i : x -→ x i are the canonical inert maps decomposing x into its components x i ∈ O 1 .

MonCat lax,R O MonCat opl,L O (1,2)-op Cocart lax,R (O) Cart opl,L (O op ) (1,2)-op . ∼ Adj ∼
At the level of objects, note that the functor Adj sends a cocartesian fibration over O to the cartesian fibration over O op classifying the same functor O -→ Cat. In particular, a cocartesian fibration over O satisfies the Segal conditions if and only if its image under Adj does. It remains to verify that the functor Adj sends a map g : C ⊗ -→ D ⊗ that preserves cocartesian lifts of inert maps to a functor F : D ⊗ -→ C ⊗ of cartesian fibrations over O op that preserves cartesian lifts of inert maps (for the reverse implication, reverse the roles of f and g in the next argument). By Proposition 3.2.7 3.2.7, this comes down to the following assertion: for any inert map β : x -→ x in O, the lax O-monoidal functor g defines the commuting left square

C ⊗ (x) D ⊗ (x) C ⊗ (x) D ⊗ (x) C ⊗ (x ) D ⊗ (x ) C ⊗ (x ) D ⊗ (x ) β ! gx β ! β ! β ! fx g x f x
and we have to verify that the associated Beck-Chevalley transformation on the right is an equivalence. Using the Segal condition, these squares can be identified with

i∈I C ⊗ (x i ) i∈I D ⊗ (x i ) i∈I C ⊗ (x i ) i∈I D ⊗ (x i ) j∈J C ⊗ (x j ) j∈J D ⊗ (x j ) j∈J C ⊗ (x j ) j∈J D ⊗ (x j ) pr (gx i ) pr pr pr (fx i ) (gx j ) (fx j )
where the vertical functors are projections associated to an inclusion of finite sets J ⊆ I. But for such projections, the Beck-Chevalley transformation is always an equivalence (since the unit and counit maps can be computed in each factor).

By considering morphism categories in the statement of Theorem 3.4.7 3.4.7, we find: Recalling

E 1 ⊗ E n E n+1 for 1 ≤ n ≤ ∞, see [Lu14 Lu14
, Theorem 5.1.2.2], we obtain:

3.4.10. Corollary. If C is a closed E n+1 -monoidal ∞-category for some 1 ≤ n ≤ ∞, then the mapping object functor [-, -] : C op × C → C carries a canonical lax E n -monoidal refinement.
For n = ∞ such a lax symmetric monoidal refinement was first established by Hinich in [Hi15 Hi15, Section A.5] using different means; his construction most certainly agrees with ours, but let us refrain from attempting a formal comparison in this paper.

Proof of Proposition 3.4.9 3.4.9. Let C ⊗ → O be the cocartesian fibration of operads witnessing the O-monoidal structure of C, and similarly for C op .

We follow the same strategy as in Example 3.1.17 3.1.17, and so wish to construct the morphism [-,-])

(C op ) ⊗ × O C ⊗ (C op ) ⊗ × O C ⊗ (C op ) ⊗ pr 1 (pr 1 ,
pr 1 in MonCat lax,R (C op ) ⊗ from its counterpart in MonCat opl,L C ⊗
using Theorem 3.4.7 3.4.7. Essentially by definition of the tensor product of operads we can regard C as an E 1 -monoid in MonCat O , so in particular the tensor product determined by the E 1 -structure is itself a (strongly) O-monoidal functor ⊗ : C × C -→ C. Applying cartesian unstraightening, we obtain a map

µ : C ⊗ × O C ⊗ -→ C ⊗ ,
with which we form

C ⊗ × O C ⊗ C ⊗ × O C ⊗ C ⊗ .
pr 1

(pr 1 ,µ)

pr 1
By Lemma 3.4.5 3.4.5 this indeed defines a morphism in MonCat opl,L (C op ) ⊗ , which dualises as desired by Corollary 3.3.16 3.3.16.

Parametrised units and counits

Consider two symmetric monoidal ∞-categories C ⊗ and D ⊗ and a lax symmetric monoidal right adjoint g : C ⊗ → D ⊗ , with left adjoint f . Given any finite collection of objects {y i } in D, we have a canonical comparison map y i → g f (y i ) given by

y i gf y i g f (y i ) , η g(µ)
where η is the unit of the adjunction f g and µ is given by the oplax monoidal structure of f . This is the prototypical example of the parametrised unit morphism that we will consider in this section.

The goal of §4.1 4.1 is to make explicit the functoriality of these maps, and in §4.2 4.2 we similarly produce a functor extracting adjoint morphisms in a parametrised adjunction. One can also obtain η β (y) from the Beck-Chevalley transformation (Definition 3.2.6 3.2.6) as

η β (y) : β * (y) β * g b f b (y) g b β * f b (y). ρ β (f b (y))
The goal of this section is to describe the fuctoriality of this unit morphism η β (y) in β and y. To motivate the functoriality in β, let us consider the following: After some preliminaries we will produce the functors η and in Construction 4.1.7 4.1.7 below. We will refer to these functors as the parametrised unit and counit, respectively.

To prepare the construction, let us write

p = (p 1 , p 2 ) : X -→ B × [1], q = (q 1 , q 2 ) : X ∨ -→ [1] × B op
for the curved orthofibrations classified by f : D -→ C and g : C ∨ -→ D ∨ respectively. Recall from Theorem 3.1.11 3.1.11 that q = (q 2 , q 1 ) is the Gray fibration dual to the curved orthofibration p. In particular,

X 0 D, X 1 C, X ∨ 0 D ∨ and X ∨ 1 C ∨ .
Naively, one could try to imitate Construction 3.2.3 3.2.3 and construct the unit map using a cocartesian pushforward of the fibre inclusion X 0 → X, followed by a cartesian pullback. More precisely, since p 2 is a cocartesian fibration, we can form the cocartesian pushforward (Construction 3.2.1 3.2.1) i 0,cc : X 0 × [1] X along p 2 of the fibre inclusion i 0 : X 0 → X; this takes y ∈ X 0,b D b to the cocartesian morphism y -→ f b (y). Dually, q 1 is a cartesian fibration and we can form the cartesian pullback

j 1,ct : X ∨ 1 × [1] X ∨
along q 1 of the fibre inclusion j 1 :

X ∨ 1 → X ∨ ; this takes x ∈ X ∨ b,1 C ∨ b to the cartesian morphism g b (x) -→ x.
To construct the unit as in Construction 3.2.3 3.2.3, we would now like take the cartesian transport of i 0,cc (and dually the cocartesian transport of j 1,ct for the counit). This can be done fibrewise over b ∈ B, but for a global construction we will need to replace p : X -→ B × [1] by its dual q : X ∨ -→ [1] × B, which is a cartesian fibration over [1]. Here we run into a problem, however: i 0,cc is a functor between curved orthofibrations which generally does not preserve cartesian arrows in the B-direction and hence does not induce a map between the dual fibrations. Indeed, for β : b -→ b and a cartesian morphism β : β * y -→ y in X 0 , the image of the cartesian arrow ( β, 1) in

X 0 × 1 under i 0,cc is f b (β * y) -→ f b (y).
This is cartesian for all β and y if and only if f : C -→ D preserves cartesian morphisms over B.

To deal with this issue (and the dual issue for the counit map), we will first extend the functor i 0,cc to the free cartesian fibration on X 0 × [1] and then dualise over B. Let us therefore briefly recall the description of free fibrations from [GHN17 GHN17, Section 4]. where the pullback is formed along evaluation at 0 and the map to B is given by evaluation at 1. Dually, define

F ct B (E) := E × B Ar(B) B; (e, b → φ(e)) d,
where the pullback is formed along evaluation at 1 and the map to B is given by evaluation at 0.

We will need the following result from [GHN17 GHN17, Theorem 4.5]:

4.1.4. Theorem. The natural maps

E F ct B (E), E F cc B (E)
over B induced by the constant diagram functor B -→ Ar(B), exhibit F ct B (E) and F cc B (E) as the free cartesian and cocartesian fibrations on φ : E -→ B, respectively. In other words, the functors Viewing X 0 × [1] as an ∞-category over B via the functor

F ct B : Cat /B Cart(B), F cc B : Cat /B Cocart (
X 0 × [1] X 0 B,
we can then extend i 0,cc to the free cartesian fibration over B as in Remark 4.1.5 4.1.5. Similarly we can extend j 1,ct to the free cocartesian fibration over B op , giving

ı 0,cc : F ct B X 0 × [1] F ct B (X 0 ) × [1] X,  1,ct : F cc B op X ∨ 1 × [1] F cc B op (X ∨ 1 ) × [1] X ∨ .
These can also be viewed as functors into arrow ∞-categories, informally given by

F ct B (X 0 ) Ar(X); y ∈ X b ,0 , b β → b β * y → β * f b y , F cc B op (X ∨ 1 ) Ar(X ∨ ); x ∈ X ∨ b,1 , b β → b β op ! g b x → β op ! x .
By construction the functors ı 0,cc and  1,ct preserve cartesian morphisms over B and cocartesian morphisms over B op , respectively. Therefore they induce functors between the dual fibrations, and we obtain functors

D cc (ı 0,cc ) : D cc F ct B (X 0 ) × [1] D cc F ct B (X 0 ) × [1] X ∨ , D ct ( 1,ct ) : D ct F cc B op (X ∨ 1 ) × [1] D ct F cc B op (X ∨ 1 ) × [1] X.
Let us note that the domains of the functors D cc (ı 0,cc ) and D ct ( 1,ct ) admit a substantial simplification, by means of the following extension of the duality between arrow and twisted arrow categories from [Ha21 Ha21, Lemma 3.1.3]:

4.1.6. Lemma. For φ : E -→ B, the duals of the free fibrations on φ can be identified as

D cc F ct B (E) E × B Tw (B) → B op , D ct F cc B (E) B × B Tw r (B) → B op naturally in φ.
We now have all the ingredients to construct the parametrised unit and counit from Theorem 4.1.2 4.1.2: 4.1.7. Construction. As before, let p = (p 1 , p 2 ) : X -→ B × [1] and q = (q 1 , q 2 ) : X ∨ -→ [1] × B op be the orthofibrations classified by f and g. Lemma 4.1.6 4.1.6 now implies that the functors ı 0,cc and  1,ct have duals

D cc (ı 0,cc ) : X 0 × B Tw (B) × [1] X ∨ , D ct ( 1,ct ) : X ∨ 1 × B op Tw r (B op ) × [1] X
that preserve cocartesian and cartesian morphisms over B, respectively. Now we can form the cartesian pullback of D cc (ı 0,cc ) via q 1 and the cocartesian pushforward of D ct ( 1,ct ) via p 2 , respectively. This gives functors

(D cc (ı 0,cc )) ct : X 0 × B Tw (B) × [1] × [1] X ∨ , (D ct ( 1,ct )) cc : X ∨ 1 × B op Tw r (B op ) × [1] × [1] X.
We can informally describe these functors as follows: the value of (D cc (ı 0,cc )) ct at an object (y ∈ X b ,0 , β : b → b ) is the square

β op ! (y) g b β op ! f b y β op ! (y) β op ! f b y in X ∨ . Note that the top horizontal arrow takes values in X ∨ 0 D ∨ . Dually, the value of (D ct ( 1,ct )) cc at (x ∈ X ∨ b ,1 , β : b → b ) is the square β * ! g b x β * x f b β * g b x β * x,
whose bottom arrow is contained in X 1 C. We then obtain the desired parametrised unit and counit maps as the restrictions

η := (D cc (ı 0,cc )) ct | X0× B Tw (B)×[1]×{0} : X 0 × B Tw (B) × [1] X ∨ 0 := (D ct ( 1,ct )) cc | X ∨ 1 × B op Tw r (B op )×[1]×{1} : X ∨ 1 × B op Tw r (B op ) × [1] X 1 .
Note that this construction is natural in X (and hence in f : C -→ D) and is compatible with base change along B -→ B. In the case where B = * is a point, the free fibration and dualisation functors are naturally equivalent to the identity [To05 To05, BGN18 BGN18] and the above construction reduces to the construction of the (co)unit from Construction 3.2.3 3.2.3. Here ev 1 is a cocartesian fibration, so we can extend η to the free cocartesian fibration on f , giving a commuting square

F cc C (D) Ar(D) C D. η ev1 g
Unwinding definitions, we find that η takes (d, f

F cc C ∨ (D 0 × B Tw (B)) Ar(D ∨ ) C ∨ D ∨ . η ev1 g
Here η is given by the assignment

y ∈ D b , b β -→ b , β * f b (y) φ -→ x -→ β * y η β (y) -→ g b β * f b y g(φ)
-→ g b x where x ∈ C ∨ b . We can also pass to the dual cartesian fibrations, which gives a commutative square (4.2.3)

(D × B Tw (B)) × C ∨ Tw r (C ∨ ) Tw r (D ∨ ) (C op ) ∨ (D op ) ∨ . η ∨ g op
y → g f (y, b ), b → g(x, b) → g(x, b ) in Tw r (D). For a morphism      y 0 b 0 b 0 f (y 0 , b 0 ) x 0 b 0 b 0 y 1 , b 1 b 1 , F (y 1 , b 1 ) x 1 , b 1 b 1      we get in Tw r (D) a morphism y 0 y 1 g(x 0 , b 0 ) g(x 1 , b 1 ).
We note that this is an equivalence if the maps y 0 → y 1 , x 1 → x 0 , and b 0 → b 1 are equivalences. This means our functor factors through the localisation of the ∞-category

(D × Tw (B)) × C×B op Tw r (C × B op )
at these morphisms. Our final goal in this section is to identify this localisation, for which we first recall a result of Hinich:

(φ) ! x → x is in W p(x ) .
Proof. This is a special case of [Hi16 Hi16, Proposition 2.1.4] (or more precisely, of the stronger result that is actually proved in [Hi16 Hi16, Section 2.2]). See also [NS18 NS18, Proposition A.14] for a generalisation, as well as a more invariant proof. This allows us to prove the following: 4.2.6. Corollary. Suppose p : E → B is a cocartesian fibration; then the identity map of E induces (via the free cocartesian fibration) a morphism of cocartesian fibrations F cc B (E) = E × B Ar(B) → E; passing to the dual cartesian fibrations we get a morphism of cartesian fibrations

E × B Tw r (B) Φ → E ∨ over B op . For any functor A → B op the induced morphism of cartesian fibrations E × B Tw r (B) × B op A Φ → E ∨ × B op A exhibits E ∨ × B op A as a localisation.
Proof. Suppose first that A → B is the identity. At the fibre over b ∈ B op we get the functor Restricting to the fibre over x ∈ C, we see in particular: 4.2.9. Corollary. In the situation of Corollary 4.2.8 4.2.8, for every x ∈ C there is a natural map

E × B B /b → E b taking (x ∈ E b , b β → b) to β ! x.
(D × B) × C C /x → Tw r (D)
sending (y, b, f (y, b) → x) to the adjoint map y → g(b, x). where the right hand (∞, 2)-categories consist of functors B → Cat as objects, (op)lax natural transformations morphisms and modifications between these as 2-morphisms. Following [START_REF] Haugseng | On lax transformations, adjunctions, and monads in (∞, 2)-categories[END_REF] we define these categories as right adjoints to the (oplax) Gray tensor product for (∞, 2)-categories constructed by Gagna, Lanari and Harpaz in [GHL21 GHL21], so that there are equivalences

Map Cat2 (A B, Cat) Map Cat2 A, Fun lax (B, Cat) Map Cat2 (B A, Cat) Map Cat2 A, Fun opl (B, Cat) .
Their Gray tensor product is defined using Lurie's scaled simplicial sets from [START_REF]∞, 2)-categories and the Goodwillie calculus I[END_REF]] as a model for (∞, 2)-categories, and so we begin with a short review of these in §5.1 5.1. In §5.2 5.2 we then show that Lurie's straightening equivalence for locally cocartesian fibrations restricts to an equivalence Fun(A B, Cat) Gray(A, B), from which we then deduce Theorem E E and Corollary F F in §5.3 5.3.

5.1. Scaled simplicial sets as a model for (∞, 2)-categories. We start by recalling a few definitions:

5.1.1. Definition. A marked simplicial set is a pair (X, T ) with X a simplicial set and T ⊆ X 1 a set of 1-simplices that contains the degenerate ones. Let sSet + denote the category of marked simplicial sets.

By [Lu09a Lu09a, Theorem 3.1.5.1] the category sSet + has a model structure Quillen equivalent to the Joyal model structure on sSet, whose fibrant objects are precisely quasicategories marked by their equivalences. We also write Cat + ∆ for the category of marked simplicial categories, i.e. categories enriched in marked simplicial sets. 5.1.2. Definition. A scaled simplicial set is a pair (X, S) with X a simplicial set and S ⊆ X 2 a set of 2-simplices that contains the degenerate ones. As usual, we will write X = (X, X 2 ) for X with the maximal scaling. Let sSet sc denote the category of scaled simplicial sets, with the morphisms being maps of simplicial sets that preserve the scalings.

We write N sc : Cat + ∆ -→ sSet sc for the scaled nerve, which takes a marked simplicial category C to the coherent nerve N C of its underlying simplicial category, scaled by the set of 2-simplices ∆ 2 -→ N C corresponding to functors of simplicial categories F : C(∆ 2 ) -→ C such that the edge ∆ 1 = C(∆ 2 )(0, 2) -→ C(F (0), F (2)) is marked; here C denotes the path category functor, left adjoint to the coherent nerve N . Its upgrade to a left adjoint of N sc we denote C sc .

The following is [Lu09b Lu09b, Theorem 4.2.7]:

Theorem (Lurie).

There is a model structure on sSet sc where the cofibrations are the monomorphisms and the weak equivalences are the maps f such that C sc is a Dwyer-Kan equivalence of marked simplicial categories. Moreover, the adjunction C sc N sc is a Quillen equivalence where Cat + ∆ carries the marked Bergner model structure. 5.1.4. Remark. An explicit description of the fibrant objects in sSet sc in terms of lifting properties has been obtained by Gagna, Harpaz and Lanari in [ [START_REF] Gagna | On the equivalence of all models for (∞, 2)-categories[END_REF].

It is then a consequence of the main results of [START_REF]∞, 2)-categories and the Goodwillie calculus I[END_REF]] that the underlying ∞-category of sSet sc is equivalent to Cat 2 . Given this, a particularly simple description of the equivalence follows from work of Barwick and Schommer-Pries [BSP21 BSP21]: Let us write Θ 2 for the full subcategory of the (ordinary) category of strict 2-categories spanned by the strict 2-categories

[m] [n 1 ], . . . , [n m ] = 0 1 2 m. n1 0 n2 0 nm 0
Since these have only identities as invertible k-morphisms, we obtain a full subcategory inclusion Θ 2 → sSet sc by viewing these 2-categories as marked simplicial categories with only degenerate edges marked and then applying the scaled nerve. Now consider the functor (5.1.5)

δ 2 : ∆ × ∆ Θ 2 → sSet sc [m], [n] [m]([n], . . . , [n]).

It now follows from the main results of [BSP21 BSP21] that the derived mapping ∞-groupoids

Map h sSet sc δ 2 (-, -), (X, S) : ∆ op × ∆ op -→ Gpd. form a complete two-fold Segal ∞-groupoid, and that this assignment induces an equivalence between the ∞-category associated to the model structure on sSet sc from Theorem 5.1.3 5.1.3 and the ∞-category of complete two-fold Segal spaces. In other words: scaled simplicial sets are a model for (∞, 2)-categories. 5.1.6. Definition. We write Cat sc for the large scaled simplicial set N sc (Set +,• ∆ ), where the category Set +,• ∆ of fibrant marked simplicial sets is regarded as enriched in itself via its internal Hom.

We will use the (∞, 2)-category associated to the scaled simplicial set Cat sc as our preferred model for the (∞, 2)-category Cat of ∞-categories.

We now recall the definition of the (oplax) Gray tensor product in terms of scaled simplicial sets, as given in [GHL21 GHL21]: 5.1.7. Definition. If (X, S) and (Y, T ) are scaled simplicial sets, we define their oplax Gray tensor product (X, S) (Y, T ) = X × Y, S T to be the scaled simplicial set with underlying simplicial set X × Y , with scaling S T consisting of the 2-simplices of the forms:

• (s 1 α, τ ) with α ∈ X 1 , τ ∈ T , • (σ, s 0 β) with σ ∈ S, β ∈ Y 1 .
For simplicial sets X and Y we will abbreviate X Y to just X Y .

From [GHL21 Composition is given by concatenation of chains. Furthermore, a subchain inclusion σ ⊆ σ is marked if it is obtained by removing one x i from σ, such that either x 0 i = x 0 i+1 or x 1 i-1 = x 1 i . On the other hand, [m] st [n] can be described as the following strict 2-category [START_REF] Haugseng | On lax transformations, adjunctions, and monads in (∞, 2)-categories[END_REF]]: its objects are tuples x = (x 0 , x 1 ) with 0 ≤ x 0 ≤ m and 0 ≤ x 1 ≤ n and Map [m] st[n] (x, y) = MaxCh x,y is the poset whose objects are maximal nondegenerate chains from x to y, with order generated by ≤ in the picture (5.1.10 5.1.10). Composition is concatenation of such chains. For each tuple x and y, we will specify a map of posets max : Ch x,y -→ MaxCh x,y as follows: for any chain σ from x to y in the grid (5.1.10 5.1.10), let σ ⊆ max(σ) be the unique maximal chain extending σ that is maximal with respect to the partial ordering on MaxCh: this means that every arrow in the chain σ going r steps right and d steps down is replaced by the maximal chain first going r steps right and then d steps down. One easily verifies that max is a map of posets, which sends every marked arrow in Ch x,y to the identity. Furthermore, it is compatible with concatenation of chains. We therefore obtain a natural map

φ : C sc (∆[m] ∆[m] ) [m] st [n]
where we view [m] st [n] as a marked simplicial category by taking nerves of mapping categories and marking equivalences (which in this case are just identities). To see that this is an equivalence, it remains to verify that max : Ch x,y -→ MaxCh x,y exhibits MaxCh x,y as the localisation of Ch x,y at the marked arrows. To see this, observe that the functor max is a cocartesian fibration. For each maximal chain τ , the inverse image of τ has a maximal element (τ itself) and for every other σ in the inverse image, the inclusion of chains σ ⊆ τ is a composite of marked arrows. It follows that the fibres of max have contractible realisation, so φ is an equivalence as desired.

We will need the following observation about the functor δ 2 : 5.2.3. Theorem (Lurie). Let (X, S) be a scaled simplicial set. Then there is a left proper combinatorial marked simplicial model structure on the slice category sSet + /X (where X denotes X with all 1-simplices marked) such that the cofibrations are the monomorphisms, and an object (E, T ) p -→ X is fibrant if and only if

(1) the underlying map of simplicial sets p : E -→ X is a locally cocartesian inner fibration,

(2) T is precisely the set of locally p-cocartesian edges in E,

(3) the locally cocartesian inner fibration p is cocartesian over S.

We write sSet + (X,S) for sSet + /X equipped with this model structure. Proof. As a simplicial model category this is a special case of [Lu09b Lu09b, Theorem 3.2.6], applied to the categorical pattern (X, X 1 , S, ∅). The marked simplicial enrichment follows from [Lu09b Lu09b, Remark 3.2.26]. 5.2.4. Theorem (Lurie). If (X, S) is a scaled simplicial set, then there is a marked simplicial Quillen equivalence Str sc (X,S) : sSet + (X,S)

Fun + (C sc (X, S), sSet + ) : Un sc (X,S)

where Fun + (C sc (X, S), sSet + ) is equipped with the projective model structure. This marked simplicial Quillen equivalence induces a weak equivalence between the underlying (fibrant) marked simplicial categories of fibrant-cofibrant objects, i.e. an equivalence of (∞, 2)categories. Combining this with Proposition 5.2.1 5.2.1, we get: 5.2.5. Corollary. Given any scaled simplicial set (X, S), there is an equivalence of fibrant scaled simplicial sets Fun sc ((X, S), Cat sc ) N sc (sSet + (X,S) ) • .

5.2.6. Remark. The categories sSet + (X,S) only depend pseudonaturally in (X, S), and therefore the equivalence in Corollary 5.2.5 5.2.5 is not literally natural at the point-set level, but this can be dealt with in the same way as in the proof of the analogous statement for the usual unstraightening equivalence in [GHN17 GHN17, Corollary A.32].

Specialising to the case of a Gray tensor product of two scaled simplicial sets, we obtain the following: 5.2.7. Proposition. Straightening for locally cocartesian fibrations gives an equivalence between maps of scaled simplicial sets (X, S) (Y, T ) -→ Cat sc and locally cocartesian inner fibrations E -→ X × Y such that (1) for x ∈ X, the restriction E x -→ Y is cocartesian over T , (2) for y ∈ Y , the restriction E y -→ X is cocartesian over S,

(3) for 1-simplices α : x -→ x in X, β : y -→ y in Y , p is cocartesian over the 2-simplex (s 1 α, s 0 β).

5.2.8. Remark. Condition (3) can be rephrased as follows: for any e ∈ E x,y , if e -→ (α, id y ) ! e a locally cocartesian morphism over (α, id y ), and (β, id y ) ! e -→ (id x , β) ! (α, id y ) ! e is a locally cocartesian morphism over (id x , β), then the composite e -→ (id x , β) ! (α, id y ) ! e is locally cocartesian over (α, β).

Proof of Proposition 5.2.7 5.2.7. Corollary 5.2.5 5.2.5 implies that there is a natural equivalence of (∞, 2)categories

Fun sc ((X, S) (Y, T ), Cat sc ) N sc (sSet + ) • (X,S) (Y,T ) .

It therefore suffices to show that the fibrant objects of (sSet + ) (X,S) (Y,T ) are precisely the locally cocartesian fibrations satisfying conditions (1)-(3) above. By definition, the fibrant objects are locally cocartesian fibrations p : E -→ X × Y such that for (σ, τ ) ∈ (S × T ) opl , the pullback (σ, τ ) * E -→ [2] is a cocartesian fibration. On the other hand, conditions (1)-(3) assert that E is cocartesian over the subset of 2-simplices (S × T ) ⊆ (S × T ) opl given as follows:

• (s 2 0 x, τ ) with x ∈ X 0 and τ ∈ T . • (σ, s 2 0 y) with σ ∈ S and y ∈ Y 0 . • (s 1 α, s 0 β) with α ∈ X 1 , β ∈ Y 1 .

We claim that this already implies that p is cocartesian over every 2-simplex in (S × T ) opl . Indeed, let us show that p is cocartesian over (σ, s 0 β), for σ ∈ S of the form Note that d 2 ξ = (σ, s 0 β), while the faces d 0 ξ = (s 1 λ, s 0 β), d 1 ξ = (s 1 µ, s 0 β) and d 3 ξ = (σ, s 2 0 y) are all in (S × T ) . If p : E -→ X × Y is cocartesian over (S × T ) , a locally cocartesian arrow e -→ (µ, β) ! e can therefore be identified in turn with the following composites of locally cocartesian arrows:

• e → (µ, id y ) ! e → (id x , β) ! (µ, id y ) ! e, since p is cocartesian over d 1 ξ,

• e → (κ, id y ) ! e → (λ, id y ) ! (κ, id y ) ! e → (id x , β) ! (c, id y ) ! e, since p is cocartesian over d 3 ξ,

• e → (κ, id y ) ! e → (λ, β) ! (κ, id y ) ! e, since p is cocartesian over d 0 ξ.

The last assertion means precisely that p is cocartesian over d 2 ξ = (σ, s 0 β), as desired.

Specialising to Gray tensor products of (∞, 1)-categories, we obtain the following: 5.2.9. Corollary. Let A and B be ∞-categories. Then there is a natural equivalence of (∞, 1)categories Fun(A B, Cat) Gray(A, B).

Proof. Combine Lemma 2.4.6 2.4.6 and Proposition 5.2.7 5.2.7.

5.2.10. Remark. Similarly to (2) (2) of Remark 2.5.7 2.5.7, it is not a priori clear to us that the equivalence constructed in Corollary 5.2.9 5.2.9 restricts to the usual straightening equivalence and Cocart(A × B) Fun(A, Cocart(B)) Fun(A × B, Cat), and per construction the equivalence from Corollary 5.2.9 5.2.9 restricts to the latter of these. Again it will follow from the uniqueness results of [HHLN21 HHLN21], that these three equivalences agree. 5.3. Unstraightening of lax natural transformations and the calculus of mates. As an application of the scaled unstraightening for Gray fibrations provided by Corollary 5.2.9 5.2.9, we will now prove the main theorem of this section: In particular, this implies that the (∞, 2)-categories Cocart lax ( * ) and Cart opl ( * ) are both equivalent to Cat, as mentioned already after Definition 3.1.7 3.1.7. 5.3.2. Remark. In fact, it follows from [START_REF] Haugseng | Two-variable fibrations and ∞-categories of spans[END_REF]] that the natural equivalence of Theorem 5.3.1 5.3.1 is essentially unique.

Proof. Let us start with the lax case, the oplax case being similar.

  and monoidal adjunctions 3. Parametrised and monoidal adjunctions 19 4. Parametrised units and counits 4. Parametrised units and counits 35 5. Lax natural transformations and the calculus of mates 5. Lax natural transformations and the calculus of mates 42

  [-, -] : C op × C -→ C in a closed E n+1 -monoidal ∞-category C admits a canonical lax E n -monoidal structure, where 1 ≤ n ≤ ∞; in [Hi15 Hi15] Hinich established the case n = ∞ by different means.

  consider a morphism in Cocart lax (B), i.e. a commutative triangle

Fun( A

 A op , Cocart lax (B)) cc CrvOrtho(A, B) Fun(B, Cart opl (A)) ct , respectively, where Fun(B, Cart opl (A)) ct denotes the wide subcategory of Fun(B, Cart opl (A)) in which the morphisms are natural transformations whose components all preserve cartesian morphisms over A, and similarly for Fun(A op , Cocart lax (B)) cc .

  2.3.5. Construction. Suppose that A = [1] op and B = [1], and let us write α, β for the unique nondegenerate simplices in A and B. Consider the diagram ρ : [1] -→ Cocart lax ([1]) corresponding to the map of cocartesian fibrations (between ordinary categories)

  to such diagrams as p-interpolating diagrams. Explicitly, the p-interpolating diagram associated to x ∈ X (a,b) is given by

(

  Recall that tailed arrows denote p-cocartesian edges and tailed arrows marked by a circle denote p r -cocartesian edges.) Now by Proposition 2.1.10 2.1.10 the composition x → (α, id) ! x → (id, β) ! (α, id) ! x along the top is still locally p-cocartesian, whence there exists an essentially unique dashed arrow as indicated making the diagram commute. More formally, consider the functor ρ : [1] -→ Cocart lax ([1]) from Construction 2.3.5 2.3.5. Then the cocartesian unstraightening of ρ over [1] can be identified with the Gray fibrationQ -→ [1] × [1],where Q is the poset is the evident one, sending 11 → 11 to the identity. Then just as in Construction 2.3.5 2.3.5 evaluation at 00 induces an equivalence between the ∞-groupoid of maps of Gray fibrations Q X[1] × [1] A × Bp α×β and ιX (a,b) . 2.4.8. Definition. If p : X -→ A × B is a Gray fibration, then a morphism Q -→ X of Gray fibrations is said to be a p-interpolating diagram. A morphism in X is said to be p-interpolating if it arises as the restriction of a p-interpolating diagram to 11 -→ 11.

  ) The functor A → Cocart lax (B) obtained by straightening p over A factors through the wide subcategory Cocart(B). Proof. Condition (1) immediately implies (2). Conversely, since p is a Gray fibration it follows from (2) that locally p-cocartesian morphisms in X are closed under composition: Given morphisms (α, β) : (a, b) → (a , b ) and (α , β ) : (a , b ) → (a , b ) in A × B, we must show that for x 00 ∈ E (a,b)

2. 5 .

 5 Dualisation of curved orthofibrations and Gray fibrations. In the present section we put together the pieces and analyse the dualisation equivalence promised in Theorem D D. 2.5.1. Theorem. Cocartesian straightening followed by cartesian unstraightening over A provides a natural equivalence (2.5.2) D ct : Gray(A, B) CrvOrtho(A op , B) : D cc between Gray fibrations over A × B and curved orthofibrations over A op × B. It restricts to the identity if A = * and is the usual dualisation equivalence between cartesian and cocartesian fibrations if B = * . In particular, for all (a, b) ∈ A × B there are canonical equivalences (2.5.3) D ct (p) (a,b) X (a,b) and D cc (q) (a,b) Y (a,b) for every Gray fibration p : X → A × B and curved orthofibration q : Y → A op × B. Dually, there is an equivalence (2.5.4) D ct : CrvOrtho(A, B) OpGray(A, B op ) : D cc with the analogous properties. Proof. Combine the straightening equivalence of Corollary 2.4.3 2.4.3 with the first equivalence of Corollary 2.3.4 2.3.4 to obtain Gray(A, B) Fun(A, Cocart lax (B)) cc CrvOrtho(A op , B). The addenda are all immediate from the construction, and the dual case is obtained by using the the equivalence from Corollary 2.3.4 2.3.4 combined with the dual of Corollary 2.4.3 2.4.3. 2.5.5. Proposition. Let p : X -→ A × B be a Gray fibration and let q : Y -→ A op × B be the dual curved orthofibration. For each α : a -→ a , β : b -→ b and x ∈ X a ,b Y a ,b , the canonical equivalence X a,b Y a,b identifies the associated p-interpolating morphism from Definition 2.4.8 2.4.8 with the associated q-interpolating morphism from Definition 2.3.8 2.3.8. Proof. The statement immediately reduces to the case where A = B = [1]. Now by construction the Gray fibration Q

  2.5.7. Remark. (1) Equivalences as on the right were first constructed by Stevenson, by comparing both Bifib(A, B) and LFib(A op × B) to an ∞-category of correspondences [St18 St18, Theorems C & D]. In the companion paper [HHLN21 HHLN21] we will prove a uniqueness result for the equivalences above that in particular shows that our equivalences restrict to those of Stevenson.

( 3 )

 3 By restricting to one of the two legs in the previous point, the dualisation of bifibrations is also discussed in detail in [HLAS16 HLAS16, Section 5], [HMS20 HMS20, Section A.1] and [CDH+20 CDH+20, Section 7.1] (4) In [HHLN21 HHLN21] we also supply a more explicit description of the equivalences in Theorem 2.5.1 2.5.1 based on span ∞-categories, generalising the work of Barwick, Glasman and Nardin [BGN18 BGN18] in the single-variable case. As a typical example of the dualisation procedure above, consider the bifibration (s, t) : Ar(C) → C × C. Its duals are the twisted arrow categories of C; let us briefly recall these to fix conventions. 2.5.8. Notation. For an ∞-category C, we write Tw (C) and Tw r (C) for the left and right twisted arrow ∞-category of C. These are characterised by the natural equivalences Map [n], Tw r (C) Map [n] [n] op , C , Map [n], Tw (C) Map [n] op [n], C , so that Tw r (C) = Tw (C) op . The natural inclusions of [n] and [n] op correspond to functors (s, t) : Tw (C) -→ C op × C, (s, t) : Tw r (C) -→ C × C op , which are a left fibration and a right fibration, respectively, both straightening to the mapping functor Map C : C op × C → Gpd. 2.5.9. Remark. Informally, the objects of Tw r (C) are the morphisms in C. For morphisms f : x → y, f : x → y in C, a morphism from f to f in Tw r (C) is a commutative diagram 10. Example. There are canonical equivalences D cc (Ar(C) → C × C) Tw (C) → C op × C and D ct (Ar(C) → C × C) Tw r (C) → C × C op . This is proved for example in [HMS20 HMS20, Corollary A.2.5], based on Lurie's recognition criterion for twisted arrow categories [Lu09a Lu09a, Corollary 5.2.1.2]. We supply another proof in [HHLN21 HHLN21] and also extend the statement to the (op)lax arrow and twisted arrow categories of an (∞, 2)-category.

  3.1. Parametrised adjunctions. We start by considering adjunctions in families over a base ∞-category B: 3.1.1. Definition. A map g : C -→ D in Cocart lax (B) is said to be a B-parametrised right adjoint if it induces right adjoint functors between the fibres over each b ∈ B. Dually, a map f : D -→ C in Cart opl (B) is said to be a B-parametrised left adjoint if it induces left adjoint functors between the fibres. Let us write Cocart lax,R (B) and Cart opl,L (B) for the wide subcategories of Cocart lax (B) and Cart opl (B) whose maps are B-parametrised right and left adjoints, respectively.

  It remains only to observe that for n = 0 the simplicial space ιF lc ([m]) is indeed constant: the unique map [m] → [0] is the localisation [m] → |[m]| * and so we know that the map ιF lc ([0]) → ιF lc ([m]) is an equivalence; since [0] is terminal in ∆ the diagram is then necessarily constant. 3.1.4. Remark. If we regard a simplicial ∞-category X : ∆ op → Cat that satisfies the Segal condition as a double ∞-category whose objects are the objects of X 0 , horizontal morphisms are the morphisms in X 0 , vertical morphisms are the objects of X 1 , and squares are the morphisms in X 1 , then the construction of Proposition 3.1.3 3.1.3 can be interpreted as extracting an (∞, 2)-category from the double ∞-category [n] → F ([n]) by forgetting the non-invertible vertical morphisms. Such a construction can be performed more generally, but the conditions in 3.1.3 3.1.3 seem required to ensure the resulting 2-fold Segal space is complete. Returning to our specific situation, for any ∞-category A we have natural equivalences Map Cat (A, Cocart lax (B × S) ιCrvOrtho(A op , B × S) Map Cat (B × S, Cart opl (A) by Corollary 2.3.4 2.3.4. By the Yoneda lemma, this implies that for all B, the functor Cat op → Cat, (B, S) → Cocart lax B × S preserves limits. Moreover, on underlying ∞-groupoids we have equivalences ι Cocart lax B × S ι Cocart B × S Map(B × S, Cat) Map(S, Fun(B, Cat)), so that the functor Cocart lax (B × |S|) → Cocart lax (B × S) corresponds to the functor Map(|S|, Fun(B, Cat)) → Map(S, Fun(B, Cat)) given by composition with S → |S|; this is therefore a monomorphism by the universal property of the localisation |S|, which says that Fun(|S|, X) → Fun(S, X) is fully faithful with image those functors that take all morphisms in S to equivalences. Let us denote by Cocart lax S B × S ⊆ Cocart lax B × S the full subcategory of cocartesian fibrations which are locally constant on S, i.e. those obtained by pulling back a cocartesian fibration over B × |S|, or equivalently those whose straightening to a functor B × S → Cat factors through the localisation to B × |S|. Applying Proposition 3.1.3 3.1.3 we then have: 3.1.5. Corollary. The functor Cat op Cat; S Cocart lax S B × S preserves limits, and the bisimplicial space

  the special case where B = * , the equivalent 2-fold complete Segal spaces Cocart lax ( * ) Cart opl ( * ) provide a model for the (∞, 2)-category Cat of ∞-categories (this is proved more precisely in Section 5.3 5.3). Consequently, we can identify Cocart lax,R ( * ) Cat R and Cart opl,L ( * ) Cat L . 3.1.9. Observation. If the ∞-category S has contractible realisation (i.e. |S| * ) then the objects of Cocart lax S B × S are by definition the cocartesian fibrations over B × S that are pulled back along the projection B × S → B, i.e. those of the form E × S → B × S for a cocartesian fibration E → B. A morphism between two such objects can then be identified with a commutative triangle E × S E B for cocartesian fibrations E, E → B. Note that this applies in particular for S = [n]. In particular, a 2-morphism in Cocart lax (B) is simply a natural transformation µ over B of maps g, g between cocartesian fibrations X → B and Y → B. This we can view as a family of natural transformations µ b : g b -→ g b that commutes with the lax structure maps, in the sense that for each b -→ b , there is a commuting diagram Depicting this diagram cubically, it can also be viewed as a lax natural transformation between two functors B × [1] -→ Cat that are constant along the interval. Note that Cocart lax,R (B) ⊆ Cocart lax (B) is the 1-full sub-2-category whose morphisms are lax natural transformations consisting of right adjoints. 3.1.10. Remark. Note that for any two ∞-categories B and S, taking opposite ∞-categories defines an equivalence (-) op : Cocart lax S (B × S) ∼ -→ Cart opl S op B op × S op . Using this, one deduces that taking opposite ∞-categories defines an equivalence of (∞, 2)-categories, where in the target the 2-morphisms are reversed (-) op : Cocart lax (B) ∼ -→ Cart opl B op 2-op .

  More generally, let us write M S (A, B × S) ⊆ M (A, B × S) and N S (A, B × S) ⊆ N (A, B × S) for the natural subspaces spanned by fibrations p : X -→ A × (B × S) such that each X a,b -→ S is locally constant, i.e. the associated functor factors through |S|. 3.1.14. Corollary. For any ∞-category B, unstraightening over [m] provides natural equivalence of 2-fold complete Segal spaces

  Example. A two-variable adjunction consists of functors F : B ×C → D and G : B op ×D → C, together with a natural equivalence Map D (F (b, c), d) Map C (c, G(b, d));

3. 2

 2 .2. Example. Let p : X -→ [1] be a cocartesian fibration classifying a functor f : D -→ C. Taking φ to be the fibre inclusion i 0 : D X 0 → X, one obtains a diagram i 0,cc : D × [1] → X. The restriction to D × {1} gives a functor D -→ X 1 C naturally equivalent to f (as a consequence of [Lu09a Lu09a, Lemma 5.2.1.4]) and for each d ∈ D, the arrow i 0,cc (d) : d f (d) is p-cocartesian. 3.2.3. Construction. Let p : X -→ [1] be a cartesian and cocartesian fibration classifying an adjoint pair f : D C : g. Applying Example 3.2.2 3.2.2 and taking the cartesian pullback of the resulting

  3.2.4. Construction. Let g : C -→ D be a morphism in Cocart lax (B) and β : b -→ b one in B. Then g determines a natural transformation of the form (

  2.1 3.2.1 and consider the diagram Fun(C(b), C) Fun(C(b), D) Fun(C(b), B). g * whose vertical maps are cocartesian fibrations. Applying the previous construction to the map β : const b ⇒ const b in the base and the fibre inclusion C(b) → C covering its domain const b , we obtain the desired natural transformation ρ β : β ! g ⇒ gβ ! . This restricts to the interpolating maps ρ β (x) defined above because g * -cocartesian arrows are given pointwise by g-cocartesian arrows (see [Lu09a Lu09a, Proposition 3.1.2.1]). When g : C -→ D is a B-parametrised right adjoint, the lax commuting square (3.2.5 3.2.5) gives rise to a natural transformation between the fibrewise left adjoints and the change-of-fibre functors β ! : 3.2.6. Definition. Consider a lax commuting square of the form (3.2.5 3.2.5) such that the horizontal functors are part of adjunctions f : D(b) C(b) : g and f : D(b ) C(b ) : g. Then the Beck-Chevalley transformation associated to ρ β is the composition

Proof.

  The equivalence Adj : Cocart lax,R (B) -→ Cart opl,L (B) is given at the level of morphisms by (3.1.16 3.1.16) for A = [1] op . In other words, consider a map g : C -→ D in Cocart lax,R (B) and let p = (p 1 , p 2 ) : X -→ [1] × B be the corresponding curved orthofibration, as in Lemma 3.1.13 3.1.13. Then the map f = Adj(g) : D -→ C in Cart(B op ) is the straightening of the curved orthofibration which is dual (relative to B) to the Gray fibration p = (p 2 , p 1 ) : X -→ B × [1]. Let us now fix β : b -→ b in B. For x ∈ C(b) X 1,b , Construction 3.2.4 3.2.4 and Example 2.

  2.4 3.2.4, replacing g : C -→ D by g : Fun(D(b), C) -→ Fun(D(b), D) and applying the above argument to the case where y is replaced by the fibre inclusion ι : D(b) → D, viewed as an object of Fun(D(b), D). Since the equivalences in Corollary 3.1.14 3.1.14 and Lemma 3.1.15 3.1.15 commute with taking functor categories (since by adjunction they commute with products), it follows that λ β is naturally equivalent to the Beck-Chevalley transformation associated to ρ β , as asserted.

  3.3 3.3.3, left us first observe that if E → [1] is the bicartesian fibration corresponding to an adjunction, then we can also phrase (3.3.1 3.3.1) in terms of the correspondence associated to this functor, in the following sense: 3.3.4. Definition. A correspondence is a left fibration X → A op × B. We define the ∞-category Corr of correspondences by the pullback Corr LFib Cat × Cat Cat, ev1 (-) op ×(-)

3. 3

 3 .10. Corollary. For any functor p : E → B, a morphism from x → y to x → y in Tw (E) of the form x x y y is Tw (p)-cartesian if x → x is p-cocartesian and y → y is p-cartesian. In particular, if p : E → B is a cartesian fibration, then Tw (E) → Tw (B) has cartesian lifts of morphisms of the form

3. 3

 3 .11. Corollary. Suppose p : E → A × B is a curved orthofibration. Then Tw B (E) → Tw (A) has cartesian lifts of morphisms in Tw (A) of the form

3. 3

 3 .16. Corollary. Let F : X × B → Y be a functor such that F (-, b) is a left adjoint for all b ∈ B, and let G : Y × B op → X be the functor corresponding to the B op -parametrised right adjoint of F , regarded as a functor X × B → Y × B over B. Then there is an equivalence

  3.4.1. Definition. The (∞, 2)-category MonCat lax O of O-monoidal ∞-categories and lax Omonoidal functors between them is given by the 1-full sub-2-category of Cocart lax (O) whose: (1) objects are O-monoidal ∞-categories, (2) morphisms are functors C ⊗ -→ D ⊗ over O that preserve the cocartesian morphisms lying over inert morphisms in O. By definition, the underlying ∞-category of MonCat lax O is the full subcategory of ∞-operads over O whose objects are the O-monoidal ∞-categories. A strong O-monoidal functor corresponds to a morphism C ⊗ -→ D ⊗ that preserves all cocartesian edges. 3.4.2. Example. Let us explicitly mention the special case O = Fin * , where MonCat lax

  3.4.3. Definition. The (∞, 2)-category MonCat opl O of O-monoidal ∞-categories and oplax Omonoidal functors between them is the sub-2-category of Cart opl (O op ) whose: (1) objects are cartesian fibrations C ⊗ → O op corresponding to O-monoids, (2) morphisms are functors C ⊗ -→ D ⊗ that preserve cartesian morphisms lying over inert morphisms in O op .

3.4. 6 .

 6 Definition. A lax O-monoidal functor g : C ⊗ -→ D ⊗ is a lax O-monoidal right adjoint if it satisfies the equivalent conditions of Lemma 3.4.5 3.4.5. Likewise, an oplax O-monoidal functor f : C ⊗ -→ D ⊗ is called an oplax O-monoidal left adjoint if it induces left adjoint functors between the fibres over each x ∈ O op (equivalently, all x ∈ O op 1 ). 3.4.7. Theorem. For each ∞-operad O, there is a natural equivalence of (∞, 2)-categories Adj : MonCat lax,R O MonCat opl,L O (1,2)-op ∼ between the 1-full sub-2-categories whose morphisms are lax O-monoidal right adjoints and oplax O-monoidal left adjoints. Proof. It suffices to show that the equivalence of Theorem 3.1.11 3.1.11 identifies the two relevant sub-2-categories

3.4. 8 .

 8 Corollary. Given an ∞-operad O and two O-monoidal ∞-categories C and D, taking adjoints gives a canonical equivalence between the ∞-category of oplax O-monoidal left adjoint functors C → D and the opposite of the ∞-category of lax O-monoidal right adjoint functors. As another application of our machinery we find: 3.4.9. Proposition. Let O be an ∞-operad and C an (E 1 ⊗ O)-monoidal ∞-category, such that the functor -⊗ x : C → C admits a right adjoint [x, -] : C → C for every x ∈ C, i.e. the monoidal ∞-category obtained by forgetting the O-monoidal structure is right closed. Then the mapping object functor [-, -] : C op × C → C carries a canonical lax O-monoidal structure.

4. 1 .

 1 Parametrised (co)units. Let us consider a parametrised left adjoint f over B with parametrised right adjoint g from Theorem 3.1.11 3.1.11 the remainder of this section we will again use (-) ∨ to denote the cocartesian fibration dual to a cartesian fibration to ease notation. Given any edge β : b → b in B and any y ∈ D b , the natural map λ β (y) : f b β * (y) -→ β * f b (y) (dual to Construction 3.2.4 3.2.4) is adjoint to a map η β (y) : β * (y) g b f b β * (y) g b β * f b (y).

g

  b λ β (y)

4.1. 1 .

 1 Example. Let β : b -→ b , γ : b -→ b and y ∈ D b . Then we claim that η γβ (y) g b β * (λ γ (y)) • η β (γ * (y)) and dually that η βα (y) ρ α (β * f b (y)) • α * η β (y). To see the first identification consider the diagramf b β * γ * (y) β * f b γ * (y) f b γ * (y) β * γ * f b (y) γ * f b (y) f b (y) λ β (γ * (y)) β * λγ (y) λγ (y)in C. The top row factors the image under f of the cartesian arrow β * γ * (y) -→ γ * (y) into λ β (γ * (y)), followed by a cartesian morphism (cf. Construction 3.2.4 3.2.4). Notice that the total composite along the top is the image under f of the cartesian arrow βγ * (y) -→ y, and that following the bottom gives a factorisation of this into a fibrewise followed by a cartesian morphism. Therefore we conclude that λ γβ (y) β * λ γ (y) • λ β (γ * (y)). Applying g b and precomposing with the unit of the adjoint pair (f b , g b ) gives the claim.The second identification arises from a dual analysis using the description of η α in terms of the mate ρ α .Example 4.1.1 4.1.1 indicates how the arrow η β depends on β via both pre-and postcomposition. Our goal will now be to make this precise by proving the following: 4.1.2. Theorem. Let f : D -→ C be a parametrised left adjoint over B with parametrised right adjoint g. Then there are canonical diagramsD × B Tw (B) Ar(D ∨ ) C ∨ × B op Tw r (B op ) Ar(C) B Ar(B) B op Ar(B op ) η const constwhose restrictions to the fibre over some b in B are equivalent to the unit and counit of the adjoint pair (f b , g b ).

  4.1.3. Notation. Given a functor φ : E → B we write F cc B (E) := E × B Ar(B) B; (e, φ(e) → b ) b,

  B) are left adjoint to the forgetful functors Cart(B) -→ Cat /B and Cocart(B) -→ Cat /B . 4.1.5. Remark. Consider a commutative triangle p is a cartesian fibration. We can extend this uniquely to a diagram cartesian morphisms. Informally, the functor f is given by e, b β -→ φ(e) -→ β * f (e), where β * f (e) → f (e) is a cartesian morphism in E over β.

4. 2 .

 2 Passing to adjoint morphisms. Next we consider the functoriality of passing to the adjoint of a morphism in the parametrised setting. We first sketch a construction in the non-parametrised case, which will have the benefit of generalising readily. Given an adjunction f : D C : g, the unit transformation η fits in a commutative square

φ

  → c) to the composite d → gf (d) g(φ) -→ g(c), i.e. to the morphism adjoint to φ. We now give a parametrised version of this construction: 4.2.1. Construction. We keep the notation of Theorem 4.1.2 4.1.2 and let f : D -→ C be a parametrised left adjoint over B, with right adjoint g : C ∨ -→ D ∨ . The parametrised unit η fits in a commutative square D × B Tw (B) Ar(D ∨ ) C ∨ D ∨ , η D cc (f ) ev1 g where D cc (f ) is obtained by first extending f : D → C to f : F ct B (D) → C and then dualising over B. At the level of objects, D cc (f ) is therefore given by y ∈ D b , β : b → b ) -→ β * f b (y). Now we can extend η over the free cocartesian fibration on D cc (f ), giving a commutative square (4.2.2)

  Construction 4.2.1 4.2.1 encodes the functoriality of passing to the adjoint morphism in the generic case of a parametrised adjunction. However if the parametrised adjunction has a particularly simple form, then the functoriality can be improved significantly: 4.2.4. Example. Recall from Example 3.1.17 3.1.17 that given a functor f :D × B → C such that for each b ∈ B, f (-, b) : D → C is a left adjoint, the diagram D × B C × B B p2 (f,id B )p2isan example of a parametrised left adjoint. In this case the parametrised unit from Theorem 4.1.2 4.1.2 is a functor η : D × Tw (B) × [1] → D. To an object (y, b β → b ) this assigns the map y → g(f (y, b ), b) adjoint to f (y, b) → f (y, b ). (y 0 , b 0 ), b 0 ) g(f (y, b 1 ), b 1 ). Now we consider the commutative square (4.2.3 4.2.3) from Construction 4.2.1 4.2.1; in our special case this simplifies to (D × Tw (B)) × C×B op Tw r (C × B op ) Tw r (D) C op × B D op . g op An object of (D × Tw (B)) × C×B op Tw r (C × B op ) can be described as a list (y, b → b , f (y, b ) → x, b → b), and the top horizontal functor takes this to the composite

4.2. 5 .

 5 Proposition (Hinich). Let p : E → B be a cocartesian fibration. Suppose for all b ∈ B we have a collection W b of morphisms in E b such that for β : b → b in B the cocartesian pushforward functor β ! : E b → E b takes W b into W b . Then we can form the cocartesian fibration E → B corresponding to the functor b → E b [W -1 b ]. The canonical morphism of cocartesian fibrations E → E exhibits E as the localisation of E at the collection of morphisms x φ → x such that p(φ) is an equivalence and p

  This has a fully faithful right adjoint (taking x ∈ E b to (x, id b )), hence it is the localisation at the class W b of morphisms (x φ → y, b γ → b β → b) such that β ! γ ! x → β ! y is an equivalence. For β : b → b ∈ B, the cartesian pullback functor (over B op ) E × B B /b → E × B B /b is given by composition with β, and hence takes W b to W b . The result then follows from (the dual of) Proposition 4.2.5 4.2.5. Finally note that the conclusion of Proposition 4.2.5 4.2.5 is preserved under base change along any functor A → B and therefore the general result follows. Taking p to be the identity of B, we obtain the following special case: 4.2.7. Corollary. For any ∞-category B, the projection Tw r (B) → B op is a localisation, as is the functor A × B op Tw r (B) → A for any functor A → B op . Returning to the B-indexed family of left adjoints f : D × B -→ C from Example 4.2.4 4.2.4, we see that the functor (D × Tw (B)) × C×B op Tw r (C × B op ) → Tw r (D) obtained from the parametrised unit factors through (D × B) × C Tw r (C). We have thus proved: 4.2.8. Corollary. Let f : D × B -→ C be a functor such that each f b : D -→ C is a left adjoint. Then there is a functor (D × B) × C Tw r (C) → Tw r (D), which takes (y, b, f (y, b) → x) to the adjoint map y → g(x, b).

  (∆[m] ∆[m]). Forgetting the marking, this simplicial category is the Boardman-Vogt resolution of [m] × [n] (cf. [MT10 MT10, Proposition 6.3.3]). Consequently, it can be identified with the simplicial category whose objects are tuples x = (x 0 , x 1 ) with 0 ≤ x 0 ≤ m and 0 ≤ x 1 ≤ n, and whereMap C sc (∆[m] ∆[n] ) (x, y) = N Ch x,yis the nerve of the poset Ch x,y of nondegenerate chains σ= [x = x 0 < x 1 < • • • < x t = y] in [m] × [n]starting at x and ending at y, ordered by subchain inclusions.

5. 1

 1 .11. Lemma. For each [m], [n] ∈ ∆, there is a natural map of (∞, 2)-categories [m] [n] -→ δ 2 [m], [n] = [m]([n], . . . , [n]) which exhibits the codomain as the localisation of [m] [n] at all 1-morphisms contained in some {i} [n]. Proof. Since [m] [n] and δ 2 ([m], [n]) = [m]([n], . . . , [n]) are both gaunt 2-categories (i.e. the only invertible 2-morphisms are the identities), the desired natural map [n] [m] -→ [m]([n], . . . , [n]) is simply the evident map of strict 2-categories that collapses all {i} [n] to the i-th vertex in [m]([n], . . . , [n]). For instance, for [m] = [2] and [n] = [1], it is given pictorially by the map collapsing the vertical 1-morphisms -→ To see that this is a localisation, note that both the domain and codomain are functors ∆ × ∆ -→ Cat 2 satisfying the co-Segal conditions; it therefore suffices to show this when [n] and [m] are 0 or 1, where the result is easily verified. 5.2. Scaled unstraightening of Gray fibrations. Let us now recall Lurie's straightening theorem for locally cocartesian fibrations over scaled simplicial sets. 5.2.1. Proposition. If C is a marked simplicial category, then the marked simplicial category Fun + (C, sSet + ) • of fibrant-cofibrant objects in the projective model structure on the enriched functor category Fun + (C, sSet + ) is weakly equivalent to Fun sc (N sc (C), Cat sc ), where Fun sc (-, -) denotes the internal Hom in scaled simplicial sets. In other words, the projective model structure on Fun + (C, sSet + ) describes the (∞, 2)-category of functors from C to Cat. Proof. This follows from [Lu09a Lu09a, Proposition A.3.4.13] since sSet + is an excellent model category by [Lu09a Lu09a, Example A.3.2.22].5.2.2. Definition. If (X, S) is a scaled simplicial set and p : E -→ X is a locally cocartesian inner fibration, then we say that p is cocartesian over S if for every σ : [2] -→ X in S, the base change σ * E -→ [2] is a cocartesian inner fibration.

Proof.

  As an (unenriched) Quillen equivalence this follows from [Lu09b Lu09b, Theorem 3.8.1]. The compatibility with the simplicial enrichment is discussed in [Lu09b Lu09b, Remark 3.8.2], and the same argument clearly extends to show that this is a marked simplicial adjunction.

  and β : y -→ y in Y 1 ; the case of a 2-simplex (s 1 α, τ ) will follow from a similar argument. Consider the 3-simplex ξ = (s 2 σ, s 2 0 β), which may be depicted as: (x , y)

Fun( A

 A × B, Cat) Cocart(A × B) : Besides this direct equivalence one can use the two inclusions Cocart(A × B) ⊆ RCocart(A, B), LCocart(A, B) and apply straightening in one factor after the other to obtain two more equivalences Cocart(A × B) Fun(B, Cocart(A)) Fun(A × B, Cat)

  5.3.1. Theorem. There are equivalences of (∞, 2)-categories Cocart lax (B) Fun lax (B, Cat) and Cart opl (B) Fun opl (B op , Cat), which are natural in B.

.

  It is a pleasure to thank Dustin Clausen, Yonatan Harpaz, Gijsbert Scheltus Karel Sebastiaan Heuts, Corina Keller, Achim Krause, Markus Land, Thomas Nikolaus, Denis Nardin, Stefan Schwede, Wolfgang Steimle and Lior Yanovski for several very useful discussions. FH and SL are members of the Hausdorff Center for Mathematics at the University of Bonn funded by the German Research Foundation (DFG), grant no. EXC 2047. FH and JN were further supported by the European Research Council (ERC) through the grants 'Moduli spaces, Manifolds and Arithmetic', grant no. 682922, and 'Derived Symplectic Geometry and Applications', grant no. 768679, respectively.

  2.1.11. Corollary. Suppose p : E → B is a locally cartesian fibration. Then p is a cartesian fibration if and only if any composite of locally p-cartesian morphisms is locally p-cartesian.

  Definition. We say that a functor p : X → A × B is cocartesian over the left factor, or simply cocartesian over A when no confusion can arise, if it satisfies the equivalent conditions of Corollary 2.2.2 2.2.2. Dually, we say that p is cartesian over A if p op is cocartesian over A op . We write LCocart(A, B) and LCart(A, B) for the subcategories of Cat/(A × B) whose objects are (co)cartesian over A, with the morphisms required to preserve the (co)cartesian morphisms over A × ιB. Similarly, we write RCocart(A, B) and RCart(A, B) for the subcategories of Cat/(A × B) whose objects are (co)cartesian over the right factor B, with the morphisms required to preserve the (co)cartesian morphisms over ιA × B.

		) is immediate from Proposition 2.2.1 2.2.1, while that of (2) and
	(3) amounts to the observation that the cocartesian morphisms for pr 1 : A × B → A are precisely
	those morphisms that project to equivalences in B.
	2.2.3. Of course, we obtain equivalences
		RCocart(A, B) LCocart(B, A),	RCart(A, B) LCart(B, A)
	by restricting the obvious equivalence Cat/(A × B) Cat/(B × A).
	From the third condition in Corollary 2.2.2 2.2.2 we immediately see:
	2.2.4. Corollary. We write pr 1 : A×B → A for the projection to A. The equivalence Cat/(A×B)
	(Cat/A)/pr 1 restricts to equivalences of subcategories
	(2.2.5)	LCocart(A, B) Cocart(A)/pr 1 ,	LCart(A, B) Cart(A)/pr 1 .
	Combining these equivalences with straightening over A, we get natural equivalences
	(2.2.6)	LCocart(A, B) Fun(A, Cat/B),	LCart(A, B) Fun(A op , Cat/B),
	since pr

1 : A × B → A straightens over A to the constant functor with value B.

  Notation. Given a functor p : X → A × B, we define p and p r by the cartesian squares Definition. A curved orthofibration is a functor of ∞-categories p : X → A × B such that p is cartesian over A and cocartesian over B, i.e. X has all p-cartesian lifts over A × ιB and all p-cocartesian lifts over ιA×B. We write CrvOrtho(A, B) for the subcategory of Cat/(A×B) whose objects are the curved orthofibrations, with the morphisms required to preserve both cartesian morphisms over A and cocartesian morphisms over B.

	2.2.8. X	X	X r
	p	p	pr
	A × ιB	A × B	ιA × B.
	To make diagrams more readable, we will sometimes indicate a p -cartesian edge of X by x • y
	and a p r -cocartesian edge of X by x • y.
	2.2.9. Remark. From Corollary 2.2.7 2.2.7 it follows immediately that for a functor p : X → A × B the
	pullback p is a (co)cartesian fibration if and only if for every b ∈ B the map on fibres p b : X b → A
	is a (co)cartesian fibration, and similarly for p r .
	2.3. Curved orthofibrations. If we combine our conditions from the previous subsection for
	a functor to A × B to straighten contravariantly over A and covariantly over B, we obtain the
	following definition:
	2.3.1. Let us record two alternative characterisations:
	2.3.2. Observation. Using Corollary 2.2.2 2.2.2 we can reformulate the definition of a curved orthofi-
	bration as a functor p = (p 1 , p 2 ) : X → A × B such that
	(1) p 1 is a cartesian fibration,
	(2) p 2 is a cocartesian fibration,
		HMS20, Lemma A.1.8] to the commutative triangle
	X	p	I × C
			pr 1
		I,
	to see that (2) implies (1).

  ) (2), note that by naturality of the straightening equivalence (Corollary 2.3.4 2.3.4) in B, we can reduce to the case where B = * . The assertion then becomes that a cartesian fibration classifies a diagram of ∞-categories and right adjoints if and only it is also a cocartesian fibration, which is [Lu14 Lu14, Proposition 4.7.4.17]. The equivalence between (2) (2)

and (4) (4) follows from characterisation (3) (3) of curved orthofibrations in Proposition 2.3.3 2.3.3, and finally the equivalence between (3) (3) and (4) (4) is part of 2.2.7 2.2.7.

  is the cocartesian fibration classified by F (-) op . Here Tw B (E) → (E op ) ∨ × B E is a left fibration by the dual of [Lu09a Lu09a, 2.4.2.11] and the observation that a locally cocartesian fibration with ∞-groupoid fibres is automatically a left fibration.

3.3.3. Theorem. If g : C → D is a B-parametrised right adjoint, with parametrised left adjoint f : D ∨ → C ∨ , then there is an equivalence

  is obtained by dualising in the second variable.

	Proof. Unstraightening over B we have the square
	RCocart([1], B)	corr B	Corr(B)
	∼		∼
	Fun(B, Cat/[1])	Fun(B,corr)	Fun(B, Corr),

so the claim follows from Theorem 3.3.5 3.3.5.

3.3.8. Proposition. Suppose p : E → [1] × B is a curved orthofibration, corresponding to a functor g : E 1 → E 0 over B. Then there is a pullback square corr

  is the curved orthofibration for f op . Combining Proposition 3.3.8 3.3.8 and Corollary 3.3.15 3.3.15, we thus get equivalences

  Tw r (C) → Tw r (C), taking (x, y, x ⊗ y → z) to the adjoint map x → [y, z]. Fixing z ∈ C, this specialises as in Corollary 4.2.9 4.2.9 to a natural functor(C × C) × C C /z → Tw r (C)that sends (x, y, x ⊗ y → z) to the adjoint morphism x → [y, z].5. Lax natural transformations and the calculus of matesThe goal of this final section is to prove Theorem E E, i.e. to produce straightening equivalencesCocart lax (B) Fun lax (B, Cat) and Cart opl (B) Fun opl (B op , Cat),

	4.2.10. Example. Let C be a closed symmetric monoidal ∞-category, with the tensor product
	viewed as a C-parametrised left adjoint as in Example 3.1.17 3.1.17. From Corollary 4.2.8 4.2.8 we obtain a
	functor
	(C × C) × C

  It follows that the oplax Gray tensor product induces a functor on the level of ∞-categories--: Cat 2 × Cat 2 -→ Cat 2 ,which preserves colimits in each variable. As the name suggests, this is supposed to be thought of as a homotopy-coherent refinement of the standard oplax Gray tensor product for strict 2-categories[START_REF] Gray | Formal category theory: adjointness for 2-categories[END_REF]]. This is supported by the following: 5.1.9. Proposition. For any m, n ≥ 0, there is a natural isomorphism between the oplax Gray tensor product [m] [n] from Theorem 5.1.8 5.1.8 and the standard oplax Gray tensor product [m] st [n] of [m] and [n], computed in strict 2-categories and depicted informally as Note that ∆[n] is a scaled simplicial set model for [n], viewed as an (∞, 2)-category. The oplax Gray tensor product from Theorem 5.1.8 5.1.8 can then be modeled by the marked simplicial category C sc

		00	10	20	m0
	(5.1.10)	01	11	21	m1
		0n	1n	2n	mn.
	Proof.

GHL21, Theorem 2.14] we quote: 5.1.8. Theorem (Gagna, Harpaz, Lanari). The oplax Gray tensor product : sSet sc × sSet sc -→ sSet sc is a left Quillen bifunctor.

Lax case. Let B be an ∞-category and consider the (∞, 2)-category Fun lax (B, Cat) determined by the natural equivalence of ∞-groupoids Map Cat2 A, Fun lax (B, Cat) Map Cat2 A B, Cat .

In terms of Segal ∞-groupoids, Fun lax (B, Cat) is then described by the bisimplicial ∞-groupoid

where δ 2 is the functor (5.1.5 5.1.5). On the other hand, the (∞, 2)-category Cocart lax (B) was defined as the complete Segal ∞-groupoid whose value on ([m], [n]) is given by the ∞-groupoid of functors

). Our goal will be to prove that there is a natural equivalence between these two bisimplicial ∞-groupoids.

To see this, let us first consider the following two natural subgroupoid inclusions:

Un sc

Un cc

The first inclusion uses the localisation The first four conditions describe the image of Un sc , by a twofold application of Proposition 5.2.7 5.2.7. The fifth condition follows from Corollary 5.2.9 5.2.9 and Lemma 5.1.11 5.1.11, together with the fact that the Gray tensor product preserves colimits in each variable, so that

On the other hand, unraveling Definition 3.1.7 3.1.7 shows that the image of the second map consists, after permuting B and [n], of functors p with the following properties:

[n] and B respectively, p is cocartesian over (s 1 κ, s 0 φ, s 0 β). as above and consider the 3-simplex ξ = s 2 s 1 (κ), s 1 s 0 (φ), s 2 0 (β) given by (i , j, b)

Unraveling the definitions as in the lax case, one sees that this is the ∞-groupoid of those functors p such that:

• denoting by pr B the projection onto B op , we have that p defines a map pr B • p -→ pr B in Cocart(B op ).

• Dualising over B op , i.e. applying cocartesian unstraightening and cartesian straightening over B, this is identified with the ∞-groupoid of maps q : where the left-hand side denotes the mapping ∞-category in Fun lax (B, Cat). Taking G to be the constant functor with value X, we get a natural equivalence Nat lax (F, const X ) Fun(Un cc (F ), X), since we have Un cc (const X ) X × B. In other words, Un cc (F ) has the universal property of the lax colimit of F : it corepresents the functor Nat lax (F, const (-) ) : Cat → Cat.

Similarly, the cartesian unstraightening of F : B → Cat is the oplax colimit: it satisfies Nat opl (F, const X ) Fun(Un ct (F ), X).

Such a characterisation of the unstraightening was first established in [ [START_REF] Gepner | Lax colimits and free fibrations in ∞-categories[END_REF], where the authors defined lax (co)limits for functors F : B → Cat as certain weighted (co)limits. As an applicaton of Theorem 5.3.1 5.3.1 we can therefore deduce that their lax colimits really have the desired universal property expressed above. Institutt for matematiske fag, NTNU Trondheim, Norway Email address: rune.haugseng@ntnu.no Mathematisches Institut, WWU Münster, Germany Email address: f.hebestreit@uni-muenster.de

Mathematisches Institut, RFWU Bonn, Germany Email address: linskens@math.uni-bonn.de IMT, Université de Toulouse III, France Email address: joost.nuiten@math.univ-toulouse.fr