Kerodon

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Example 8.2.3.5. Let $\operatorname{\mathcal{C}}$ be a (locally small) category. Then 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 $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. More precisely, there is a natural transformation $\alpha : \underline{\Delta ^0}_{\operatorname{Tw}(\operatorname{\mathcal{C}})} \rightarrow \mathscr {H}|_{ \operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}))}$ which exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, given explicitly by assigning to each object $(f: X \rightarrow Y)$ of $\operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$ the inclusion map $\{ f\} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) = \mathscr {H}(X,Y)$.