Warning 9.2.2.2. In the formulation of Definition 9.2.2.1, we have implicitly assumed that the $\infty $-category $\operatorname{\mathcal{C}}$ is locally small (so that the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \bullet ): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is well-defined). More generally, let $\mu $ be a regular cardinal which is not small such that $\operatorname{\mathcal{C}}$ is locally $\mu $-small. In this case, we say that $C \in \operatorname{\mathcal{C}}$ is compact if the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \bullet ): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is finitary. It follows from Corollary 7.4.3.8 that this condition does not depend on the choice of $\mu $.
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