Kerodon

Corollary 8.6.9.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration with covariant transport representation $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$, and let $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be a right fibration with covariant transport representation $\mathscr {F}': \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$. Then a Kan complex $X$ is a coend $\operatorname{Coend}( \mathscr {F}, \mathscr {F}' )$ (in the $\infty $-category $\operatorname{\mathcal{S}}$) if and only if it is weakly homotopy equivalent to the $\infty $-category $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$.