Theorem 8.6.9.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration with covariant transport representation $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, and let $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration with contravariant transport representation $\mathscr {F}': \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$. Let $W$ be the collection of morphisms $(w, w')$ of $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ such that $w$ is $U$-cocartesian and $w'$ is $U'$-cartesian. Then the localization $( \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}' )[W^{-1} ]$ is a coend $\operatorname{Coend}( \mathscr {F}, \mathscr {F}' )$.
Proof of Theorem 8.6.9.9. Using Corollary 8.6.2.4, we can choose a cartesian fibration $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and a map $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ which exhibits $U^{\dagger }$ as a cartesian conjugate of $U$. Then $\mathscr {F}$ is also a contravariant transport representation for $U^{\dagger }$. Set $\operatorname{\mathcal{M}}= \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$, so that projection onto the middle factor determines a cartesian fibration $V: \operatorname{\mathcal{M}}\rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$ whose contravariant transport representation identifies with the functor
Let $\widetilde{W}$ be the collection of all $V$-cartesian morphisms of $\operatorname{\mathcal{M}}$: that is, morphisms of $\operatorname{\mathcal{M}}$ whose image in $\operatorname{\mathcal{E}}^{\dagger }$ is $U^{\dagger }$-cartesian and whose image in $\operatorname{\mathcal{E}}'$ is $U'$-cartesian. Our assumption on $T$ guarantees that the induced map $T': \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ carries $\widetilde{W}$ into $W$. Moreover, we have the following partial converse:
- $(\ast )$
Let $(w,w'): (X,X') \rightarrow (Y,Y')$ be a morphism in $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ which belongs to $W$. Then $(w,w')$ is isomorphic to $T'( \widetilde{w} )$, for some morphism $\widetilde{w} \in \widetilde{W}$.
To prove $(\ast )$, let $f: C \rightarrow D$ be the morphism of $\operatorname{\mathcal{C}}$ given by the common image of $w$ and $w'$. Since $T$ induces an equivalence of $\infty $-categories $T_{C}: \operatorname{\mathcal{E}}^{\dagger }_{C} \rightarrow \operatorname{\mathcal{E}}_{C}$, there exists an object $X^{\dagger } \in \operatorname{\mathcal{E}}^{\dagger }_{C}$ such that $T_ C( X^{\dagger } )$ is isomorphic to $X$. Let $u$ be the morphism of $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ given by $( \operatorname{id}_{ X^{\dagger } }, f_{L})$, where $f_{L}: \operatorname{id}_{C} \rightarrow f$ is the morphism in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ described in Example 8.1.3.6. Our assumption on $T$ guarantees that $T(u)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$ whose source is equal to $f$, and is therefore isomorphic to $w$ as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \Delta ^1, \operatorname{\mathcal{E}})$. It follows that $\widetilde{w} = (u, w' )$ is a morphism of $\operatorname{\mathcal{M}}$ which satisfies the requirement of $(\ast )$.
Let $\widetilde{W}_{0}$ be the subset of $\widetilde{W}$ consisting of those morphisms $\widetilde{w}$ of $\operatorname{\mathcal{M}}$ such that the image of $\widetilde{w}$ in $\operatorname{\mathcal{E}}^{\dagger }$ is $U^{\dagger }$-cartesian and the image of $\widetilde{w}$ in $\operatorname{\mathcal{E}}'$ is degenerate. It follows from Proposition 8.6.9.1 that the functor $T'$ exhibits $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ as a localization of $\operatorname{\mathcal{M}}$ with respect to $\widetilde{W}$. Combining this observation with $(\ast )$ and Proposition 6.3.1.22, we conclude that $( \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}' )[ W^{-1} ]$ is a localization of $\operatorname{\mathcal{M}}$ with respect to $\widetilde{W}$, which we can identify with the coend $\operatorname{Coend}( \mathscr {F}, \mathscr {F}' )$ (see Proposition 8.6.8.15). $\square$