# Kerodon

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Corollary 3.5.1.10. 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.

Proof. Since colimits in the category of simplicial sets are computed levelwise (Remark 1.1.1.13), 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 3.5.1.9, since the product $\Delta ^ n \times X$ is also a finite simplicial set (Remark 3.5.1.6 and Example 3.5.1.2. $\square$
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