Corollary 3.5.4.13. Let $n$ be an integer and let $K$ and $X$ be simplicial sets. If $X$ is weakly $n$-coskeletal, then $\operatorname{Fun}(K,X)$ is also weakly $n$-coskeletal.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. It follows from Corollary 3.5.3.12 that $\operatorname{Fun}(K,X)$ is $(n+1)$-coskeletal. It will therefore suffice to show that, if $n \geq -1$, then the restriction map $\theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{n+1}, \operatorname{Fun}(K,X) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{n+1}, \operatorname{Fun}(K, X) )$ is injective. Note that $\theta $ can be identified with the restriction map
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{n+1} \times K, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{n+1} \times K, X ). \]
Since $\operatorname{\partial \Delta }^{n+1} \times K$ contains the $n$-skeleton of $\Delta ^{n+1} \times K$, the injectivity of this map follows from Proposition 3.5.4.12. $\square$