Kerodon

Proof. Fix an uncountable cardinal $\mu $ having cofinality $\geq \lambda $ and exponential cofinality $\geq \kappa $, and such that $\operatorname{\mathcal{C}}$ is locally $\mu $-small. Let $\operatorname{Fun}^{( \kappa , \lambda )-\operatorname{fin}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ spanned by the $(\kappa , \lambda )$-finitary functors. Then the subcategory $\operatorname{Fun}^{( \kappa , \lambda )-\operatorname{fin}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ is closed under $\kappa $-small limits (Remark 9.1.9.9). By definition, an object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact if and only if the contravariant Yoneda embedding $h^{\bullet }: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ carries $C$ to an object of $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$. The desired result now follows from the observation that $h^{\bullet }$ preserves $\kappa $-small limits (see Proposition 7.4.1.18). $\square$