Corollary 3.6.1.10. Let $X$ be a finite simplicial set. Then the functor
\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad Y \mapsto \operatorname{Fun}(X,Y) \]
commutes with filtered colimits.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Since colimits in the category of simplicial sets are computed levelwise (Remark 1.1.0.8), it will suffice to that the functor
\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set}\quad \quad Y \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, \operatorname{Fun}(X,Y) ) \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n \times X, Y ) \]
commutes with filtered colimits for each $n \geq 0$. This is a special case of Proposition 3.6.1.9, since the product $\Delta ^ n \times X$ is also a finite simplicial set (Remark 3.6.1.6 and Example 3.6.1.2. $\square$