Corollary 9.3.6.7 (0691)

Corollary 9.3.6.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa \trianglelefteq \lambda \trianglelefteq \mu $ be regular cardinals, where $\lambda $ is uncountable. Then an object $X \in \operatorname{Ind}^{\mu }_{\kappa }(\operatorname{\mathcal{C}})$ is $(\lambda ,\mu )$-compact if and only if it belongs to the full subcategory $\operatorname{Ind}^{\lambda }_{\kappa }(\operatorname{\mathcal{C}}) \subseteq \operatorname{Ind}^{\mu }_{\kappa }(\operatorname{\mathcal{C}})$ (see Remark 9.3.6.1).

Proof. Since $\lambda $ is uncountable, the $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is idempotent-complete (Proposition 9.2.1.14). The desired result now follows by combining Theorem 9.3.6.4 with Corollary 9.3.2.8. $\square$