Corollary 11.10.2.13. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cartesian fibration of categories which is equipped with a cleavage $(f,Y) \mapsto \widetilde{f}_{Y}$, and let $\chi _{U}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ be the transport representation of Construction 11.10.2.4. Then $\chi _{U}$ is a functor of $2$-categories if and only if $U$ is a fibration of categories.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$