Remark 5.5.4.9 (Functoriality of the Core). For every $\infty $-category $\operatorname{\mathcal{C}}$, let $\operatorname{\mathcal{C}}^{\simeq }$ denote the core of $\operatorname{\mathcal{C}}$ (Construction 4.4.3.1). The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\simeq }$ determines a simplicially enriched functor $\operatorname{QCat}\rightarrow \operatorname{Kan}$. Passing to homotopy coherent nerves, we obtain a functor of $\infty $-categories
\[ \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{S}}\quad \quad \operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\simeq }. \]