Kerodon

Proof. Fix a small regular cardinal $\kappa $ for which $\operatorname{\mathcal{C}}$ admits small $\kappa $-filtered colimits (this condition is satisfied, for example, if $\operatorname{\mathcal{C}}$ is accessible). Using Proposition 9.4.6.16, we can choose a small regular cardinal $\lambda > \kappa $ such that $K$ is $\lambda $-small and $\operatorname{\mathcal{C}}$ is $\lambda $-accessible. We will complete the proof by showing that $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$ is $\lambda $-accessible. Let $\operatorname{\mathcal{C}}_{< \lambda }$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $\lambda $-compact objects. Since $\operatorname{\mathcal{C}}_{< \lambda }$ is $\kappa $-sequentially cocomplete, Corollary 9.4.5.14 implies that the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is $\lambda $-compactly generated, and that the full subcategory of $\lambda $-compact objects coincides with $\operatorname{Fun}(K, \operatorname{\mathcal{C}}_{< \lambda } )$. To complete the proof, it will suffice to show that the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}}_{< \lambda } )$ is essentially small. This follows from Remark 4.9.5.13, since $\operatorname{\mathcal{C}}_{< \lambda }$ and $K$ are essentially small. $\square$