Corollary 4.7.4.12. Let $\kappa $ be an uncountable cardinal, let $S$ be a $\kappa $-small simplicial set, and let $K$ be a finite simplicial set. Then the simplicial set $\operatorname{Fun}(K, S)$ is $\kappa $-small.
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Proof. Without loss of generality, we may assume that $\kappa $ is the least uncountable cardinal for which $S$ is $\kappa $-small. In particular, $\kappa $ is regular (Remark 4.7.4.7). By virtue of Proposition 4.7.4.9, it will suffice to show that for every finite simplicial set $L$, the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( L, \operatorname{Fun}(K, S) ) \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K \times L, S)$ is $\kappa $-small. This is a special case of Proposition 4.7.4.9, since the simplicial set $K \times L$ is finite (Remark 3.6.1.6). $\square$