Proposition 9.2.2.29 (Transitivity). Let $\kappa \trianglelefteq \lambda \trianglelefteq \mu $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is $(\kappa ,\mu )$-cocomplete. The following conditions are equivalent:
The functor $F$ is $(\kappa ,\mu )$-finitary.
The functor $F$ is $(\kappa ,\lambda )$-finitary and $(\lambda ,\mu )$-finitary.
Proof. We will assume that $\kappa < \lambda $ (otherwise, there is nothing to prove). The implication $(1) \Rightarrow (2)$ follows from Remark 9.2.2.8 (and requires only the assumption $\kappa \leq \lambda \leq \mu $). The converse 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). $\square$