Corollary 4.7.4.19. Let $\lambda $ be an uncountable cardinal and let $\kappa = \mathrm{ecf}(\lambda )$ be its exponential cofinality. If $S$ is a $\lambda $-small simplicial set and $K$ be a $\kappa $-small simplicial set. Then $\operatorname{Fun}(K,S)$ is $\lambda $-small.
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Proof. Since $K$ is $\kappa $-small, we can choose an epimorphism of simplicial sets ${\coprod }_{i \in I} \Delta ^{n_{i}} \twoheadrightarrow K$, where $I$ is a $\kappa $-small set. It follows that $\operatorname{Fun}(K,S)$ can be identified with a simplicial subset of the product ${\prod }_{ i \in I} \operatorname{Fun}( \Delta ^{n_ i}, S)$. Corollary 4.7.4.14 guarantees that each factor $\operatorname{Fun}( \Delta ^{n_ i}, S)$ is $\lambda $-small, so that the product ${\prod }_{ i \in I} \operatorname{Fun}( \Delta ^{n_ i}, S)$ is $\lambda $-small by virtue of Corollary 4.7.4.18. $\square$