Kerodon

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Proposition 2.4.6.21. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $U: \operatorname{\mathcal{C}}_{\bullet } \rightarrow ( \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} )_{\bullet }$ be the simplicial functor described in Remark 2.4.6.19. Then, for any strict $2$-category $\operatorname{\mathcal{D}}$, composition with $U$ induces a bijection category $\operatorname{\mathcal{D}}$, composition with $U$ induces a bijection

\[ \{ \textnormal{Strict functors $f: \mathrm{h}_{2} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{D}}$} \} \rightarrow \{ \textnormal{Simplicial Functors $F: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \operatorname{\mathcal{D}}_{\bullet }$} \} ; \]

here $\operatorname{\mathcal{D}}_{\bullet }$ denote the simplicial category associated to $\operatorname{\mathcal{D}}$ by Example 2.4.2.7.