Definition 9.2.2.12. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits. We say that an object $C \in \operatorname{\mathcal{C}}$ is $(\kappa , \lambda )$-compact if the corepresentable functor
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, \bullet ): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu } \quad \quad D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D) \]
is $(\kappa ,\lambda )$-finitary: that is, it preserves $\lambda $-small $\kappa $-filtered colimits. Here $\mu $ is any cardinal of cofinality $\geq \lambda $ such that $\operatorname{\mathcal{C}}$ is locally $\mu $-small (it follows from Corollary 7.4.3.8 that this condition does not depend on the choice of $\mu $).