Corollary 2.4.5.10. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category, let $0 < i < n$, and let $\sigma _0: \Lambda ^{n}_{i} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ be a map of simplicial sets, which we can identify with a simplicial functor $F: \operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$ inducing a map of simplicial sets
\[ \lambda _0: \boldsymbol {\sqcap }^{n-1}_{i} \simeq \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^ n_ i] }( 0, n)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }. \]
Then we have a canonical bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Maps $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ with $\sigma _0 = \sigma |_{ \Lambda ^ n_{i}}$} \} \ar [d] \\ \{ \textnormal{Maps $\lambda : \operatorname{\raise {0.1ex}{\square }}^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }$ with $\lambda _0 = \lambda |_{ \boldsymbol {\sqcap }^{n-1}_{i}}$} \} .} \]