Construction 8.3.6.1. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. Then the construction $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ determines a simplicial functor $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$. Passing to homotopy coherent nerves, we obtain a functor of $\infty $-categories
\[ \mathscr {H}_{\operatorname{\mathcal{C}}}: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{Kan}) = \operatorname{\mathcal{S}}. \]