# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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

$\xymatrix@R =50pt@C=50pt{ \{ \textnormal{Maps \sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}) with \sigma _0 = \sigma |_{ \Lambda ^ n_{i}}} \} \ar [d] \\ \{ \textnormal{Maps \lambda : \operatorname{\raise {0.1ex}{\square }}^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet } with \lambda _0 = \lambda |_{ \boldsymbol {\sqcap }^{n-1}_{i}}} \} .}$