Example 8.3.3.4. Let $\operatorname{\mathcal{C}}$ be a category. The construction $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ determines a functor
\[ \mathscr {H}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set}) \subset \operatorname{\mathcal{S}}\quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), \]
which is a $\operatorname{Hom}$-functor for the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. For a more general statement, see Proposition 8.3.6.2.