Corollary 8.6.4.9. 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
Let $\lambda : \widetilde{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ be a left fibration which exhibits $U^{\vee }$ as a cocartesian dual of $U$. Then $\lambda $ induces an isomorphism of functors $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \xrightarrow {\sim } \operatorname{hTr}_{ \operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}}^{\operatorname{op}}$, which carries each vertex $C \in \operatorname{\mathcal{C}}$ to (the isomorphism class of) a functor which represents the balanced coupling $\lambda _{C}: \widetilde{\operatorname{\mathcal{E}}}_ C \rightarrow \operatorname{\mathcal{E}}^{\vee }_{C} \times \operatorname{\mathcal{E}}_{C}$.