Kerodon

Remark 8.6.8.9 (Representable Functors). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty $-categories and let $\mathscr {F} = \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ be its contravariant transport representation. Then the functor $\mathscr {F}$ is representable (in the sense of Variant 5.6.6.2) if and only if the $\infty $-category $\operatorname{\mathcal{E}}$ has a final object. More precisely, $\mathscr {F}$ is representable by an object $C \in \operatorname{\mathcal{C}}$ if and only if $C$ can be lifted to a final object of the $\infty $-category $\operatorname{\mathcal{E}}$. See Proposition 5.6.6.21.