Remark 5.6.5.15. Let $\operatorname{\mathcal{C}}$ be a simplicial set equipped with a functor $\overline{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$. It follows from Corollary 5.6.5.12 that the functor $\overline{\mathscr {F}}$ is isomorphic to the homotopy transport representation of a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ if and only if it can be promoted to a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$