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Proposition 5.4.6.6. The coslice comparison functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}_{\ast } ) \rightarrow \operatorname{\mathcal{QC}}_{\ast }$ is an equivalence of $\infty $-categories.

Proof. By virtue of Theorem 5.4.2.21, it will suffice to show that for every pair of objects $(\operatorname{\mathcal{C}},C), (\operatorname{\mathcal{D}}, D) \in \operatorname{\mathcal{QC}}_{\ast }$, the restriction map

\[ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq } = \operatorname{Hom}_{\operatorname{QCat}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{QCat}}( \{ C\} , \operatorname{\mathcal{D}}) = \operatorname{Fun}( \{ C\} , \operatorname{\mathcal{D}})^{\simeq } \]

is a Kan fibration. This follows from Proposition 4.4.3.6, since the restriction functor $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \{ C\} , \operatorname{\mathcal{D}})$ is an isofibration of $\infty $-categories (Corollary 4.4.5.3). $\square$