Remark 5.5.5.4 (020N): Comparison with $\operatorname{\mathcal{QC}}$

Remark 5.5.5.4 (Comparison with $\operatorname{\mathcal{QC}}$). Let $\operatorname{QCat}$ and $\operatorname{\mathbf{QCat}}$ be the simplicial categories defined in Constructions 5.5.4.1 and 5.5.5.1, respectively. There is an evident comparison map $\operatorname{QCat}\hookrightarrow \operatorname{\mathbf{QCat}}$ which is the identity at the level of objects, and which is given on morphism spaces by the inclusion maps

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

Passing to the homotopy coherent nerve, we obtain a functor of $(\infty ,2)$-categories $\operatorname{\mathcal{QC}}\hookrightarrow \operatorname{ \pmb {\mathcal{QC}} }$ which restricts to an isomorphism of $\infty $-categories $\operatorname{\mathcal{QC}}\simeq \operatorname{Pith}( \operatorname{ \pmb {\mathcal{QC}} })$ (Corollary 5.4.8.8).