Kerodon

Remark 4.7.5.10. Let $\lambda $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\lambda $-small, and let $K$ be a simplicial set. Suppose that $K$ is $\kappa $-small, where $\kappa = \mathrm{ecf}(\lambda )$ is the exponential cofinality of $\lambda $. Then the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is essentially $\lambda $-small. To prove this, we can use Remark 4.5.1.16 to reduce to the case where $\operatorname{\mathcal{C}}$ is $\lambda $-small, in which case it follows from Corollary 4.7.4.12. Moreover, if $\kappa $ is uncountable, then it suffices to assume that $K$ is essentially $\kappa $-small.