Proposition 8.6.8.11. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration between small simplicial sets. Then $\operatorname{Fun}^{\operatorname{Cart}}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is a limit of the contravariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let us identify $\operatorname{Fun}^{\operatorname{Cart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ with the opposite of the $\infty $-category $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}^{\operatorname{op}} )$ of cocartesian sections of $U^{\operatorname{op}}$. By virtue of Remark 8.6.8.7 (and the fact that the opposition functor $\sigma : \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{QC}}$ is an equivalence of $\infty $-categories), it will suffice to show that $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{E}}^{\operatorname{op}} )$ is a limit of the covariant transport representation $\operatorname{Tr}_{ \operatorname{\mathcal{E}}^{\operatorname{op}} / \operatorname{\mathcal{C}}^{\operatorname{op}} }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$, which is the content of Proposition 7.4.4.1. $\square$