Remark 3.5.4.21. Let $X$ be a simplicial set, let $n$ be an integer, and let $\operatorname{cosk}^{\circ }_{n}(X)$ denote the weak $n$-coskeleton of $X$. It follows from Proposition 3.5.4.12 that, for every simplicial set $S$, the restriction map
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(S, \operatorname{cosk}^{\circ }_{n}(X) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), \operatorname{cosk}^{\circ }_{n}(X) ) \xleftarrow {\sim } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), X) \]
is an injection, whose image consists of those morphisms $f: \operatorname{sk}_{n}(S) \rightarrow X$ which can be extended to the $(n+1)$-skeleton of $S$.