Proposition 8.6.8.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration between small simplicial sets and let $W$ be the collection of $U$-cartesian edges of $\operatorname{\mathcal{E}}$. Then $\operatorname{\mathcal{E}}[W^{-1}]$ is a colimit 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 $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be the cocartesian fibration determined by $U$ and let $W^{\operatorname{op}}$ be the collection of $U^{\operatorname{op}}$-cocartesian edges of $\operatorname{\mathcal{E}}^{\operatorname{op}}$. 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 $\operatorname{\mathcal{E}}^{\operatorname{op}}[ W^{\operatorname{op},-1} ]$ is a colimit 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 7.4.5.1. $\square$