Proposition 3.5.3.16 (Existence). Let $X$ be a simplicial set. For every integer $n$, there exists a simplicial set $\operatorname{cosk}_{n}(X)$ and a morphism $f: X \rightarrow \operatorname{cosk}_{n}(X)$ which exhibits $\operatorname{cosk}_{n}(X)$ as an $n$-coskeleton of $X$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\operatorname{cosk}_{n}(X)$ denote the simplicial set given by the construction
\[ ( [m] \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}( \Delta ^{m} ), X ), \]
and let $f: X \rightarrow \operatorname{cosk}_{n}(X)$ be the morphism of simplicial sets given on $m$-simplices by the restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}( \Delta ^ m), X) )$. If $m \leq n$, then $\operatorname{sk}_{n}( \Delta ^ m ) = \Delta ^ m$; it follows that $f$ is bijective on $m$-simplices for $m \leq n$. We will complete the proof by showing that $\operatorname{cosk}_{n}(X)$ is $n$-coskeletal. Fix an integer $m > n$; we wish to show that the restriction map
\[ \theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ m, \operatorname{cosk}_{n}(X)) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, \operatorname{cosk}_{n}(X) ) \]
is a bijection. Writing $\operatorname{\partial \Delta }^{m}$ as a colimit of simplices (Remark 1.1.3.13) and applying Corollary 1.1.4.9, we can identify $\theta $ with the restriction map
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}( \Delta ^{m} ), X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}( \operatorname{\partial \Delta }^ m), X ). \]
The desired result now follows from the observation that the $n$-skeleton of $\Delta ^ m$ is contained in $\operatorname{\partial \Delta }^ m$. $\square$