Remark 3.5.3.21. Let $X$ be a simplicial set, let $n$ be an integer, and let $\operatorname{cosk}_{n}(X)$ denote the $n$-coskeleton of $X$. For every simplicial set $S$, we have canonical isomorphisms
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), X ) \xrightarrow {\sim } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), \operatorname{cosk}_{n}(X) ) \xleftarrow {\sim } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{cosk}_{n}(X) ) \]
where the map on the left is bijective because the map $X \rightarrow \operatorname{cosk}_{n}(X)$ is bijective on simplices of dimension $\leq n$, and the map on the right is bijective by virtue of Proposition 3.5.3.10. Note that this observation was implicitly used (in the special case $S = \operatorname{\partial \Delta }^{m}$) in the proof of Proposition 3.5.3.16.