Corollary 1.1.4.9. For every integer $k$, the skeleton functor $\operatorname{sk}_{k}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ preserves small colimits.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $S: \operatorname{\mathcal{J}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets; we wish to show that the comparison map
\[ \theta : \varinjlim _{J \in \operatorname{\mathcal{J}}} \operatorname{sk}_{k}( S(J) ) \rightarrow \operatorname{sk}_{k}( \varinjlim _{J \in \operatorname{\mathcal{J}}} S(J) ) \]
is an isomorphism of simplicial sets. Using Propositions 1.1.4.6 and 1.1.3.11, we see that the source and target of $\theta $ are simplicial sets of dimension $\leq k$. It will therefore suffice to show that $\theta $ induces a bijection on $n$-simplices for $n \leq k$ (Corollary 1.1.3.14), which follows immediately from Remark 1.1.4.4 (and Remark 1.1.0.8). $\square$