Corollary 5.3.6.10. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a cocartesian fibration of simplicial sets and let
\[ \operatorname{hTr}_{\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}}: \mathrm{h} \mathit{\operatorname{\mathcal{B}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}} \]
be the homotopy transport representation for $U$ (Construction 5.2.5.2). Then, for any $\infty $-category $\operatorname{\mathcal{D}}$, the composition
\[ \mathrm{h} \mathit{\operatorname{\mathcal{B}}}^{\operatorname{op}} \xrightarrow { \operatorname{hTr}_{\operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}}^{\operatorname{op}} } \mathrm{h} \mathit{\operatorname{QCat}}^{\operatorname{op}} \xrightarrow { \operatorname{Fun}( -, \operatorname{\mathcal{D}}) } \mathrm{h} \mathit{\operatorname{QCat}} \]
is the homotopy transport representation for the cartesian fibration $\operatorname{Fun}( \operatorname{\mathcal{C}}/\operatorname{\mathcal{B}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{B}}$ of Corollary 5.3.6.8.