$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary 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.

Proof. Since colimits in the category of simplicial sets are computed levelwise (Remark, 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, since the product $\Delta ^ n \times X$ is also a finite simplicial set (Remark and Example $\square$