Corollary 4.7.8.8. Let $\kappa $ be an infinite cardinal, let $\lambda $ be an uncountable cardinal of exponential cofinality $\geq \kappa $ (Definition 4.7.3.16), and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\lambda $-small. Then, for every $\kappa $-small simplicial set $K$, the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is locally $\lambda $-small. Moreover, if $\kappa $ is uncountable, then it suffices to assume that $K$ is essentially $\kappa $-small.
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Proof. Let $F,F': K \rightarrow \operatorname{\mathcal{C}}$ be diagrams; we wish to show that the morphism space $\operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})}(F,F')$ is essentially $\lambda $-small. Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the essential images of $F$ and $F'$. Proposition 4.7.8.7 guarantees that $\operatorname{\mathcal{C}}_0$ is essentially $\lambda $-small. It will therefore suffice to show that $\operatorname{Fun}(K, \operatorname{\mathcal{C}}_0)$ is locally $\lambda $-small, which follows immediately from Remark 4.7.5.10. $\square$