Kerodon

Proposition 9.2.2.18. Let $\kappa $, $\lambda $, and $\mu $ be regular cardinals satisfying $\kappa \trianglelefteq \lambda \trianglelefteq \mu $, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\mu $-small $\kappa $-filtered colimits. Then an object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\mu )$-compact if and only if it is both $(\kappa ,\lambda )$-compact and $(\lambda ,\mu )$-compact.