Corollary 9.2.2.19. Let $\kappa $ and $\lambda $ be small regular cardinals satisfying $\kappa \trianglelefteq \lambda $, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small $\kappa $-filtered colimits. Then an object $C \in \operatorname{\mathcal{C}}$ is $\kappa $-compact if and only if it is both $(\kappa ,\lambda )$-compact and $\lambda $-compact.
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