Corollary 3.5.3.12. Let $n$ be an integer and let $K$ and $X$ be simplicial sets. If $X$ is $n$-coskeletal, then $\operatorname{Fun}(K,X)$ is also $n$-coskeletal.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Proposition 3.5.3.10, it will suffice to show that for every simplicial set $S$, the restriction map $\theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S, \operatorname{Fun}(K,X) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S), \operatorname{Fun}(K,X) )$ is a bijection. Using Proposition 1.5.3.2, we can identify $\theta $ with the horizontal map in the commutative diagram
\[ \xymatrix@R =50pt@C=30pt{ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S \times K, X) \ar [rr] \ar [dr] & & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S) \times K, X ) \ar [dl] \\ & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(S \times K), X). } \]
It will therefore suffice to show that the vertical maps in this diagram are bijections, which follows from our assumption that $X$ is $n$-coskeletal (Proposition 3.5.3.10). $\square$