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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 .