Kerodon

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Corollary 2.4.6.13. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category, let $n > 0$, and let $\sigma _0: \partial \Delta ^ n \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ be a map of simplicial sets, which we identify with a simplicial functor $F: \operatorname{Path}[ \partial \Delta ^ n ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$ inducing a map of simplicial sets

\[ \lambda _0: \operatorname{\partial \raise {0.1ex}{\square }}^{n-1} \simeq \operatorname{Hom}_{ \operatorname{Path}[ \partial \Delta ^ n ] }( 0, n)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }. \]

Then we have a canonical bijection

\[ \xymatrix { \{ \textnormal{Maps $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ with $\sigma _0 = \sigma |_{ \partial \Delta ^ n }$} \} \ar [d] \\ \{ \textnormal{Maps $\lambda : \operatorname{\raise {0.1ex}{\square }}^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }$ with $\lambda _0 = \lambda |_{ \operatorname{\partial \raise {0.1ex}{\square }}^{n-1}}$ } \} }. \]