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Proposition 5.6.6.14. Let $V: \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \rightarrow \operatorname{\mathcal{QC}}$ be the cocartesian fibration of Proposition 5.6.6.11 and let

\[ \operatorname{hTr}_{ \operatorname{\mathcal{QC}}_{\operatorname{Obj}} / \operatorname{\mathcal{QC}}}: \mathrm{h} \mathit{\operatorname{\mathcal{QC}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}} \]

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 $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched functor) to the isomorphism $\mathrm{h} \mathit{\operatorname{QCat}} \simeq \mathrm{h} \mathit{\operatorname{\mathcal{QC}}}$ supplied by Remark 5.6.4.6. In particular, $\operatorname{hTr}_{\operatorname{\mathcal{QC}}_{\operatorname{Obj}} / \operatorname{\mathcal{QC}}}$ is an equivalence of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched categories.

Proof. Apply Theorem 5.5.9.2 to the simplicial category $\operatorname{\mathbf{QCat}}$. $\square$