Remark 5.5.4.6. Let $\mathrm{h} \mathit{\operatorname{QCat}}$ denote the homotopy category of $\infty $-categories (Construction 4.5.1.1), which we view as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category (see Remark 3.1.5.12). Applying Proposition 2.4.6.9 and Corollary 4.6.9.20, we obtain a canonical isomorphism of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched categories $\Phi : \mathrm{h} \mathit{\operatorname{QCat}} \xrightarrow {\sim } \mathrm{h} \mathit{\operatorname{\mathcal{QC}}}$, which is given on objects by the construction $\Phi (\operatorname{\mathcal{C}}) = \operatorname{\mathcal{C}}$ and on morphism spaces by the homotopy equivalences
\[ \operatorname{Hom}_{\operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{QC}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \]
of Remark 5.5.4.5.