Corollary 8.6.1.7. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration, and let $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian fibration. If $U^{\dagger }$ is a cartesian conjugate of $U$, then the homotopy transport representations
\[ \operatorname{hTr}_{\operatorname{\mathcal{E}}^{\dagger } / \operatorname{\mathcal{C}}^{\operatorname{op}} }, \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}} \]
are isomorphic.