Proposition 9.2.1.15 (Transitivity). Let $\kappa $, $\lambda $, and $\mu $ be regular cardinals satisfying $\kappa \trianglelefteq \lambda \trianglelefteq \mu $. Then an $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\mu )$-cocomplete if and only if it is both $(\kappa ,\lambda )$-cocomplete and $(\lambda ,\mu )$-cocomplete.
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Proof. We will assume $\kappa < \lambda $ (otherwise, there is nothing to prove). Assume that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-cocomplete and $(\lambda ,\mu )$-cocomplete; we wish to show that it is $(\kappa ,\mu )$-cocomplete (the converse follows immediately from Remark 9.2.1.5, and requires only the assumption $\kappa \leq \lambda \leq \mu $). This follows from Corollary 9.1.6.6, since every $\mu $-small $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{K}}$ can be realized as a $\mu $-small, $\lambda $-filtered colimit of $\lambda $-small $\kappa $-filtered $\infty $-categories (Corollary 9.1.7.16). $\square$