Corollary 8.6.4.8. 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
and let $\operatorname{hTr}_{\operatorname{\mathcal{E}}^{\vee }/\operatorname{\mathcal{C}}}^{\operatorname{op}}$ denote the functor $C \mapsto \operatorname{hTr}_{\operatorname{\mathcal{E}}^{\vee }/\operatorname{\mathcal{C}}}(C)^{\operatorname{op}} = (\operatorname{\mathcal{E}}^{\vee }_{C})^{\operatorname{op}}$. Let $\lambda : \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ be a left fibration such that, for each vertex $C \in \operatorname{\mathcal{C}}$, the coupling $\lambda _{C}: \widetilde{\operatorname{\mathcal{E}}}_ C \rightarrow \operatorname{\mathcal{E}}^{\vee }_{C} \times \operatorname{\mathcal{E}}_{C}$ is representable by a functor $G_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow ( \operatorname{\mathcal{E}}^{\vee }_{C})^{\operatorname{op}}$. if $\lambda $ satisfies condition $(b)$ of Definition 8.6.4.1, then the construction $C \mapsto [G_ C]$ determines a natural transformation of functors $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \rightarrow \operatorname{hTr}_{ \operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}}^{\operatorname{op}}$.