Kerodon

Remark 5.6.5.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and let $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ be the homotopy transport representation of $U$ (Construction 5.2.5.2). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets and let $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ be the induced functor between homotopy categories. Let $\alpha : \operatorname{\mathcal{E}}\rightarrow \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ be an equivalence of cocartesian fibrations over $\operatorname{\mathcal{C}}$. By virtue of Corollary 5.6.2.22, $\alpha $ induces an isomorphism from $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ to $\mathrm{h} \mathit{\mathscr {F}}$ in the functor category $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \mathrm{h} \mathit{\operatorname{QCat}})$. Stated more informally, any covariant transport representation of $U$ provides a lifting of the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ from the ordinary category $\operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \mathrm{h} \mathit{\operatorname{QCat}} )$ to the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$. Moreover, if the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category, then the identification $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \simeq \mathrm{h} \mathit{\mathscr {F}}$ is an isomorphism of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functors (Proposition 5.6.2.20).