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8.6.7 The Opposition Functor

Recall that, for every $\infty$-category $\operatorname{\mathcal{C}}$, the opposite simplicial set $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is also an $\infty$-category (Proposition 1.4.2.6). Our goal in this section is to show that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\operatorname{op}}$ can be promoted to a functor of $\infty$-categories $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$, where $\operatorname{\mathcal{QC}}$ denotes the $\infty$-category of (small) $\infty$-categories (Construction 5.5.4.1). Beware that this is not completely obvious from the definition. The $\infty$-category $\operatorname{\mathcal{QC}}$ was obtained as the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$, where $\operatorname{QCat}$ denotes the simplicial category whose objects are $\infty$-categories and whose morphism spaces are given by the formula $\operatorname{Hom}_{\operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } = \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$. The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\operatorname{op}}$ determines an automorphism of $\operatorname{QCat}$ as an ordinary category. However, this automorphism is not compatible with the simplicial enrichment of $\operatorname{QCat}$: for $\infty$-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, the Kan complex $\operatorname{Hom}_{ \operatorname{QCat}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}} )_{\bullet } = \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}} )^{\simeq }$ identifies with the opposite of the Kan complex $\operatorname{Hom}_{ \operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } = \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$. To address this point, it is convenient to work with a slight variant of Construction 5.5.4.1.

Notation 8.6.7.1. Let $\operatorname{\mathcal{E}}$ be a simplicial category. We define a new simplicial category $\operatorname{\mathcal{E}}^{\asymp }$ as follows:

• The objects of $\operatorname{\mathcal{E}}^{\asymp }$ are the objects of $\operatorname{\mathcal{E}}$.

• For every pair of objects $X,Y \in \operatorname{\mathcal{E}}$, the simplicial set $\operatorname{Hom}_{ \operatorname{\mathcal{E}}^{\asymp } }( X, Y)_{\bullet }$ is the twisted arrow construction $\operatorname{Tw}( \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\bullet } )$.

• For every triple of objects $X,Y,Z \in \operatorname{\mathcal{E}}$, the composition law

$\circ : \operatorname{Hom}_{ \operatorname{\mathcal{E}}^{\asymp } }( Y,Z)_{\bullet } \times \operatorname{Hom}_{ \operatorname{\mathcal{E}}^{\asymp } }( X,Y )_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{E}}^{\asymp } }( X, Z)_{\bullet }$

is obtained by applying the twisted arrow functor $\operatorname{Tw}$ to the composition law for the simplicial category $\operatorname{\mathcal{E}}$.

The simplicial category $\operatorname{\mathcal{E}}^{\asymp }$ is equipped with a simplicial functor $\pi : \operatorname{\mathcal{E}}^{\asymp } \rightarrow \operatorname{\mathcal{E}}$, which carries each object to itself and is given on morphism spaces by the projection map

$\operatorname{Hom}_{ \operatorname{\mathcal{E}}^{\asymp }}(X,Y)_{\bullet } = \operatorname{Tw}( \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\bullet }$

described in Notation 8.1.1.6.

Proposition 8.6.7.2. Let $\operatorname{\mathcal{E}}$ be a locally Kan simplicial category. Then:

$(1)$

The simplicial category $\operatorname{\mathcal{E}}^{\asymp }$ of Notation 8.6.7.1 is locally Kan.

$(2)$

The forgetful functor $\pi : \operatorname{\mathcal{E}}^{\asymp } \rightarrow \operatorname{\mathcal{E}}$ is a weak equivalence of simplicial categories (see Definition 4.6.8.7).

$(3)$

The functor $\pi$ induces an equivalence of $\infty$-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{E}}^{\asymp } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}})$.

Proof. Assertions $(1)$ and $(2)$ follow immediately from Corollary 8.1.2.3; assertion $(3)$ then follows from Corollary 4.6.8.8. $\square$

Remark 8.6.7.3 (Comparison with the Conjugate). Let $\operatorname{\mathcal{E}}$ be a simplicial category. Recall that the conjugate $\operatorname{\mathcal{E}}^{\operatorname{c}}$ is a simplicial category having the same objects, with morphism spaces given by $\operatorname{Hom}_{\operatorname{\mathcal{E}}^{\operatorname{c}}}(X,Y)_{\bullet } = \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\bullet }^{\operatorname{op}}$ (see Example 2.4.2.12). Then there is a canonical isomorphism of simplicial categories $\operatorname{\mathcal{E}}^{\asymp } \xrightarrow {\sim } ( \operatorname{\mathcal{E}}^{\operatorname{c}} )^{\asymp }$, which is the identity on objects and given on morphism spaces by the isomorphisms

$\operatorname{Hom}_{\operatorname{\mathcal{E}}^{\asymp }}( X,Y )_{\bullet } = \operatorname{Tw}( \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\bullet } ) \simeq \operatorname{Tw}( \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\bullet }^{\operatorname{op}} ) = \operatorname{Hom}_{ (\operatorname{\mathcal{E}}^{\operatorname{c}} )^{\asymp } }( X,Y)_{\bullet }$

described in Remark 8.1.1.7. Composing this isomorphism with the forgetful functor $(\operatorname{\mathcal{E}}^{\operatorname{c}})^{\asymp } \rightarrow \operatorname{\mathcal{E}}^{\operatorname{c}}$, we obtain a forgetful functor $\pi ^{\operatorname{c}}: \operatorname{\mathcal{E}}^{\asymp } \rightarrow \operatorname{\mathcal{E}}^{\operatorname{c}}$. If $\operatorname{\mathcal{E}}$ is locally Kan, then Proposition 8.6.7.2 guarantees that $\pi ^{\operatorname{c}}$ is a weak equivalence of simplicial categories. We therefore obtain equivalences of $\infty$-categories

$\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{E}}) \xleftarrow {\sim } \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{E}}^{\asymp } ) \xrightarrow {\sim } \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{E}}^{c} ).$

We now specialize to the case of interest to us.

Construction 8.6.7.4. Let $\operatorname{QCat}$ denote the simplicial category whose objects are (small) $\infty$-categories, with morphisms spaces given by $\operatorname{Hom}_{ \operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ (see Construction 5.5.4.1). We let $\operatorname{QCat}^{\asymp }$ denote the simplicial category described in Notation 8.6.7.1, and we let $\operatorname{\mathcal{QC}}^{\asymp }$ denote the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}^{\asymp } )$.

Proposition 8.6.7.5. The simplicial set $\operatorname{\mathcal{QC}}^{\asymp }$ is an $\infty$-category. Moreover, the forgetful functor $\operatorname{QCat}^{\asymp } \rightarrow \operatorname{QCat}$ of Notation 8.6.7.1 induces an equivalence of $\infty$-categories $\pi : \operatorname{\mathcal{QC}}^{\asymp } \rightarrow \operatorname{\mathcal{QC}}$.

Proof. Apply Proposition 8.6.7.2 to the locally Kan simplicial category $\operatorname{\mathcal{E}}= \operatorname{QCat}$. $\square$

Construction 8.6.7.6 (The Opposite Functor). The simplicial category $\operatorname{QCat}^{\asymp }$ is equipped with an automorphism $\widetilde{\sigma }$, given on objects by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\operatorname{op}}$ and on morphism spaces by the composition

\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{QCat}^{\asymp } }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } & = & \operatorname{Tw}( \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } ) \\ & \simeq & \operatorname{Tw}( (\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } )^{\operatorname{op}} ) \\ & \simeq & \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}})^{\simeq } ) \\ & = & \operatorname{Hom}_{ \operatorname{QCat}^{\asymp } }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}} ), \end{eqnarray*}

where the isomorphism on the second line is supplied by Remark 8.1.1.7. It follows from Proposition 8.6.7.5 that there exists a functor of $\infty$-categories $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ for which the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{QC}}^{\asymp } \ar [d]^{\pi } \ar [r]^-{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \widetilde{\sigma } ) }_{\sim } & \operatorname{\mathcal{QC}}^{\asymp } \ar [d]^{\pi } \\ \operatorname{\mathcal{QC}}\ar [r]^-{\sigma } & \operatorname{\mathcal{QC}}}$

commutes up to isomorphism. Moreover, the functor $\sigma$ is uniquely deterrmined up to isomorphism. We will refer to $\sigma$ as the opposition functor for the $\infty$-category $\operatorname{\mathcal{QC}}$.

The terminology of Construction 8.6.7.6 is justified by the following observation:

Proposition 8.6.7.7. Let $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ be the opposition functor of Construction 8.6.7.6. Then the diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \operatorname{QCat}) \ar [r]^-{\sigma _0} \ar [d] & \operatorname{N}_{\bullet }( \operatorname{QCat}) \ar [d] \\ \operatorname{\mathcal{QC}}\ar [r]^-{\sigma } & \operatorname{\mathcal{QC}}}$

commutes up to isomorphism, where the functor $\sigma _0$ is given by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\operatorname{op}}$.

Proof. Let $\operatorname{QCat}^{\circ }$ denote the underlying category of the simplicial category $\operatorname{QCat}$, and let us abuse notation by viewing $\operatorname{QCat}^{\circ }$ as a constant simplicial category. We can then identify $\operatorname{QCat}^{\circ }$ with a simplicial subcategory of $\operatorname{QCat}$ having the same objects, with morphism spaces given by

$\operatorname{Hom}_{\operatorname{QCat}^{\circ }}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } = \operatorname{sk}_{0}( \operatorname{Hom}_{ \operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } ).$

Note that the inclusion map $\operatorname{QCat}^{\circ } \hookrightarrow \operatorname{QCat}$ factors as a composition

$\operatorname{QCat}^{\circ } \xrightarrow { \iota } \operatorname{QCat}^{\asymp } \xrightarrow { \pi } \operatorname{QCat},$

where the functor $\iota : \operatorname{QCat}^{\circ } \rightarrow \operatorname{QCat}^{\asymp }$ carries each $\infty$-category $\operatorname{\mathcal{C}}$ to itself and each functor of $\infty$-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ to the vertex

$\operatorname{id}_{F} \in \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } ) = \operatorname{Hom}_{ \operatorname{QCat}^{\asymp } }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet }.$

We then have a diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \operatorname{QCat}) \ar [r]^-{\sigma _0} \ar [d]^{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\iota )} & \operatorname{N}_{\bullet }( \operatorname{QCat}) \ar [d]^{\operatorname{N}_{\bullet }^{\operatorname{hc}}( \iota ) } \\ \operatorname{\mathcal{QC}}^{\asymp } \ar [d] \ar [r]^-{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \widetilde{\sigma }) } & \operatorname{\mathcal{QC}}^{\asymp } \ar [d] \\ \operatorname{\mathcal{QC}}\ar [r]^-{\sigma } & \operatorname{\mathcal{QC}}, }$

where the upper square is strictly commutative and the lower square commutes up to isomorphism. It follows that the outer rectangle also commutes up to isomorphism. $\square$

Remark 8.6.7.8. Let $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ be the opposition functor of Construction 8.6.7.6. Passing to homotopy categories, we obtain a functor $\overline{\sigma }: \mathrm{h} \mathit{\operatorname{QCat}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$. It follows from Proposition 8.6.7.7 that, up to isomorphism, $\overline{\sigma }$ agrees with the automorphism of $\mathrm{h} \mathit{\operatorname{QCat}}$ which is given on objects by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\operatorname{op}}$, and on morphisms by the construction $[F] \mapsto [F^{\operatorname{op}} ]$; here $[F]$ denotes the isomorphism class of a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $[F^{\operatorname{op}}]$ the isomorphism class of the opposition functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$.

Proposition 8.6.7.9 (Involutivity). Let $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ be the opposition functor. Then the composition $\sigma \circ \sigma$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{QC}}}$. In particular, $\sigma$ is an equivalence of $\infty$-categories.

Proof. This follows from Proposition 8.6.7.5, since the composition $\widetilde{\sigma } \circ \widetilde{\sigma }$ is equal to the identity functor of the $\infty$-category $\operatorname{\mathcal{QC}}^{\asymp }$. $\square$

Remark 8.6.7.10 (Uniqueness). Let $\operatorname{Aut}( \operatorname{\mathcal{QC}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{QC}}, \operatorname{\mathcal{QC}})$ spanned by those functors $\operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ which are equivalences of $\infty$-categories. A theorem of Toën () guarantees that $\operatorname{Aut}( \operatorname{\mathcal{QC}})$ is a Kan complex having exactly two connected components, each of which is contractible (see Corollary ). Consequently, the opposition functor $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ of Construction 8.6.7.4 is characterized (up to a contractible space of choices) by the fact that it is an equivalence of $\infty$-categories which is not isomorphic to the identity functor $\operatorname{id}_{ \operatorname{\mathcal{QC}}}$.

Recall that every Kan complex $X$ is homotopy equivalent to its opposite $X^{\operatorname{op}}$ (Example 8.6.4.18). The following is a more precise statement:

Proposition 8.6.7.11. Let $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ be the opposition functor of Construction 8.6.7.6. Then the restriction $\sigma |_{ \operatorname{\mathcal{S}}}$ is isomorphic to the identity functor from $\operatorname{\mathcal{S}}\subset \operatorname{\mathcal{QC}}$ to itself.

Proof. For every $\infty$-category $\operatorname{\mathcal{C}}$, the image $\sigma (\operatorname{\mathcal{C}}) \in \operatorname{\mathcal{QC}}$ is equivalent to the opposite $\infty$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (Remark 8.6.7.8); in particular, $\operatorname{\mathcal{C}}$ is a Kan complex if and only if $\sigma ( \operatorname{\mathcal{C}})$ is a Kan complex. It follows that $\sigma$ restrict a functor $\sigma _0: \operatorname{\mathcal{S}}\rightarrow \operatorname{\mathcal{S}}$, which is also an equivalence of $\infty$-categories. We wish to show that $\sigma _0$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{S}}}$. This follows from Example 8.4.0.4, since $\sigma _0( \Delta ^0 )$ is homotopy equivalent to the Kan complex $( \Delta ^0)^{\operatorname{op}} \simeq \Delta ^0$. $\square$

Using the classification of cocartesian fibrations given in §5.6, we can use the opposition functor $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ to give a reformulation of cocartesian duality.

Proposition 8.6.7.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets having transport representations $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}, \operatorname{Tr}_{\operatorname{\mathcal{E}}^{\vee }/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$. Then $U^{\vee }$ is a cocartesian dual of $U$ if and only if $\operatorname{Tr}_{\operatorname{\mathcal{E}}^{\vee }/\operatorname{\mathcal{C}}}$ is isomorphic to $\sigma \circ \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$, where $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ denotes the opposition functor of Construction 8.6.7.6.

Proof. Assume that $\operatorname{Tr}_{\operatorname{\mathcal{E}}^{\vee }/\operatorname{\mathcal{C}}}$ is isomorphic to $\sigma \circ \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$; we will show that $U^{\vee }$ is cocartesian dual to $U$ (the reverse implication then follows formally from the fact that cocartesian duals are unique up to equivalence; see Theorem 8.6.5.1). Let $\pi : \operatorname{QCat}^{\asymp } \rightarrow \operatorname{QCat}$ denote the forgetful functor, and let $\pi ': \operatorname{QCat}^{\asymp } \rightarrow \operatorname{QCat}$ be the composition of $\pi$ with the automorphism $\widetilde{\sigma }: \operatorname{QCat}^{\asymp } \simeq \operatorname{QCat}^{\asymp }$ described in Construction 8.6.7.6. By virtue of Proposition 8.6.7.5, we may assume without loss of generality that the covariant transport representation $\mathscr {F}_{+} = \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ factors as a composition $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\pi ) \circ \mathscr {F}$ for some diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}^{\asymp }$. Set $\mathscr {F}_{-} = \operatorname{N}_{\bullet }^{\operatorname{hc}}(\pi ') \circ \mathscr {F}$. Our assumption then guarantees that $\mathscr {F}_{-}$ is a covariant transport representation for $U^{\vee }$. We may therefore assume without loss of generality that $U$ and $U^{\vee }$ coincide with the projection maps $\int _{\operatorname{\mathcal{C}}} \mathscr {F}_{+} \rightarrow \operatorname{\mathcal{C}}$ and $\int _{\operatorname{\mathcal{C}}} \mathscr {F}_{-} \rightarrow \operatorname{\mathcal{C}}$, respectively.

We now proceed as in the proof of Proposition 8.6.4.23. Define a simplicial functor $\tau : \operatorname{QCat}^{\asymp } \rightarrow \operatorname{QCat}$ as follows:

• On objects, $\tau$ is given by the construction $\operatorname{\mathcal{D}}\mapsto \operatorname{Tw}(\operatorname{\mathcal{D}})$.

• On morphism spaces, $\tau$ is given by the morphism of simplicial sets

\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{QCat}^{\asymp } }( \operatorname{\mathcal{D}}, \operatorname{\mathcal{D}}' )_{\bullet } & = & \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}}')^{\simeq } ) \\ & \rightarrow & \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{Tw}(\operatorname{\mathcal{D}}') )^{\simeq } \\ & = & \operatorname{Hom}_{ \operatorname{QCat}}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{Tw}(\operatorname{\mathcal{D}}') )_{\bullet }. \end{eqnarray*}

which classifies the composition

$\operatorname{Tw}(\operatorname{\mathcal{D}}) \times \operatorname{Tw}( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}}')^{\simeq } ) \hookrightarrow \operatorname{Tw}(\operatorname{\mathcal{D}}\times \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}}' ) ) \xrightarrow { \operatorname{Tw}(\operatorname{ev}) } \operatorname{Tw}(\operatorname{\mathcal{D}}'),$

where $\operatorname{ev}: \operatorname{\mathcal{D}}\times \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}}' ) \rightarrow \operatorname{\mathcal{D}}'$ is the evaluation map.

Let $\widetilde{\mathscr {F}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ denote the diagram given by the composition $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \tau ) \circ \mathscr {F}$, and set $\widetilde{\operatorname{\mathcal{E}}} = \int _{\operatorname{\mathcal{C}}} \widetilde{\mathscr {F}}$. There is a natural transformation of simplicial functors $\tau \rightarrow \pi ' \times \pi$, which carries each $\infty$-category $\operatorname{\mathcal{D}}$ to the left fibration $\operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}$ of Proposition 8.1.1.11. Applying Corollary , see that this natural transformation induces a left fibration

$\widetilde{\operatorname{\mathcal{E}}} = \int _{\operatorname{\mathcal{C}}} \widetilde{\mathscr {F}} \xrightarrow {\lambda } \int _{\operatorname{\mathcal{C}}} ( \mathscr {F}_{-} \times \mathscr {F}_{+} ) \simeq \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}.$

We will complete the proof by showing that $\lambda$ exhibits $U^{\vee }$ as a cocartesian dual of $U$: that is, it satisfies conditions $(a)$ and $(b)$ of Definition 8.6.4.1.

$(a)$

Fix a vertex $C \in \operatorname{\mathcal{C}}$; we wish to show that the left fibration

$\lambda _{C}: \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \widetilde{\mathscr {F}} \rightarrow ( \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}_{-} ) \times ( \{ C \} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}_{+} )$

is a balanced coupling. Set $\operatorname{\mathcal{D}}= \widetilde{\mathscr {F}}(C)$. Using Example 5.6.2.18, we obtain a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(\operatorname{\mathcal{D}}) \ar [r] \ar [d] & \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [d]^{ \lambda _{C} } \\ \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\ar [r] & ( \{ C\} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}_{-} ) \times ( \{ C \} \times _{\operatorname{\mathcal{C}}} \int _{\operatorname{\mathcal{C}}} \mathscr {F}_{+} ), }$

where the horizontal maps are equivalences of $\infty$-categories. The desired result now follows from the observation that the twisted arrow coupling $\operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}$ is balanced (Example 8.2.6.2).

$(b)$

Let $U: \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map, let $f: X \rightarrow X'$ be a $U$-cocartesian edge of $\widetilde{\operatorname{\mathcal{E}}}$, and let $\overline{f}: C \rightarrow C'$ denote the image of $f$ in the simplicial set $\operatorname{\mathcal{C}}$. The functor $\mathscr {F}$ carries the vertex $C$ to an $\infty$-category $\operatorname{\mathcal{D}}$, $C'$ to an $\infty$-category $\operatorname{\mathcal{D}}'$, and $\overline{f}$ to a vertex of the Kan complex $\operatorname{Tw}( \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{D}}' )^{\simeq } )$, which we can identify with an isomorphism $u: F_{-} \rightarrow F_{+}$ between functors $F_{-}, F_{+}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}'$. We can then identify $X$ with a morphism $e: D_{-} \rightarrow D_{+}$ in the $\infty$-category $\operatorname{\mathcal{D}}$ and $X'$ with a morphism $e': D'_{-} \rightarrow D'_{+}$ in the $\infty$-category $\operatorname{\mathcal{D}}'$, so that $f$ determines a morphism in the $\infty$-category $\operatorname{Tw}(\operatorname{\mathcal{D}}')$ which we depict informally in the diagram $\operatorname{\mathcal{D}}'$ which we depict informally in the diagram

$\xymatrix@R =50pt@C=50pt{ F_{-}( D_{-} ) \ar [d]^{ u(e) } & D'_{-} \ar [l] \ar [d]^{ e' } \\ F_{+}( D_{+} ) \ar [r] & D'_{+}. }$

Our assumption that $X$ is universal for the coupling $\lambda _{C}$ guarantees that $e$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{D}}$ (Example 8.2.1.5), so that the left vertical map is an isomorphism in the $\infty$-category $\operatorname{\mathcal{D}}'$. Our assumption that $f$ is $U$-cocartesian guarantees that the horizontal maps in the diagram are also isomorphisms (Remark 5.6.2.14). It follows that $e'$ is also an isomorphism in $\operatorname{\mathcal{D}}$, so that $X'$ is universal for the coupling $\lambda _{C'}$ as desired.

$\square$

Remark 8.6.7.13. Let $\operatorname{\mathcal{QC}}_{\operatorname{Obj}}$ denote the $\infty$-category of pairs $(\operatorname{\mathcal{C}}, X)$, where $\operatorname{\mathcal{C}}$ is a small $\infty$-category and $X$ is an object of $\operatorname{\mathcal{C}}$ (see Definition 5.5.6.10). Then the identity functor $\operatorname{id}: \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation for the universal cocartesian fibration

$U: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}\quad \quad (\operatorname{\mathcal{C}}, X) \mapsto \operatorname{\mathcal{C}}.$

Applying Proposition 8.6.7.12, we deduce that the opposition functor $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ is a covariant transport representation for a cocartesian dual of the fibration $U$. By virtue of Corollary 5.6.5.13, this property characterizes the functor $\sigma$ up to isomorphism.