Corollary 8.6.4.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ be cocartesian fibrations of simplicial sets having homotopy transport representations
\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}, \operatorname{hTr}_{\operatorname{\mathcal{E}}^{\vee }/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}. \]
If $U^{\vee }$ is a cocartesian dual of $U$, then $\operatorname{hTr}_{\operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}}$ is isomorphic to the functor
\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}^{\operatorname{op}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}} \quad \quad C \mapsto \operatorname{\mathcal{E}}_{C}^{\operatorname{op}}. \]