Proposition 5.5.5.14. Let $V: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ be the cocartesian fibration of Proposition 5.5.5.11 and let
denote the enriched homotopy transport representation of Construction 5.2.8.9. Then $\operatorname{hTr}_{\operatorname{\mathcal{QC}}_{\operatorname{Obj}} / \operatorname{\mathcal{QC}}}$ is homotopy inverse (as an $\operatorname {h}\! \mathit{\operatorname{Kan}}$-enriched functor) to the isomorphism $\operatorname {h}\! \mathit{\operatorname{QCat}} \simeq \operatorname {h}\! \mathit{\operatorname{\mathcal{QC}}}$ supplied by Remark 5.5.3.6. In particular, $\operatorname{hTr}_{\operatorname{\mathcal{QC}}_{\operatorname{Obj}} / \operatorname{\mathcal{QC}}}$ is an equivalence of $\operatorname {h}\! \mathit{\operatorname{Kan}}$-enriched categories.