Kerodon

Proof. Let $m$ be a positive integer. Fix an integer $0 \leq i \leq m$ and a morphism of simplicial sets $\sigma _0: \Lambda ^{m}_{i} \rightarrow \operatorname{cosk}_{n}(X)$; we wish to show that $\sigma _0$ can be extended to an $m$-simplex of $\operatorname{cosk}_ n(X)$. Using Remark 3.5.3.21, we can identify $\sigma _0$ with a morphism of simplicial sets $f_0: \operatorname{sk}_{n}( \Lambda ^{m}_{i} ) \rightarrow X$; we wish to show that $f_0$ can be extended to the $n$-skeleton of $\Delta ^{m}$. If $n < m-1$, then $\operatorname{sk}_{n}( \Lambda ^{m}_{i} ) = \operatorname{sk}_{n}( \Delta ^{m} )$ and there is nothing to prove. We may therefore assume that $n \geq m-1$, so that $\operatorname{sk}_{n}( \Lambda ^{m}_{i} ) = \Lambda ^{m}_{i}$. In this case, our assumption that $X$ is a Kan complex guarantees that $f_0$ can be extended to an $n$-simplex of $X$. $\square$