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Proposition 2.4.6.7. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $u: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \underline{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }_{\bullet }$ be the simplicial functor described in Remark 2.4.6.2. Then, for any category $\operatorname{\mathcal{D}}$, composition with $u$ induces a bijection

\[ \{ \textnormal{Ordinary Functors $f: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$} \} \rightarrow \{ \textnormal{Simplicial Functors $F: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \underline{\operatorname{\mathcal{D}}}_{\bullet }$} \} . \]

Proof. Use Proposition 1.1.6.19. $\square$