Proposition 9.1.9.19 (062T): Transitivity

Proposition 9.1.9.19 (Transitivity). Let $\kappa $, $\lambda $, and $\mu $ be regular cardinals satisfying $\kappa \trianglelefteq \lambda $ and $\lambda \trianglelefteq \mu $. For every $\infty $-category $\operatorname{\mathcal{C}}$, the following conditions are equivalent:

$(1)$: The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\mu $-small, $\kappa $-filtered colimits.
$(2)$: The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\mu $-small, $\lambda $-filtered colimits and $\lambda $-small, $\kappa $-filtered colimits.

Moreover, if these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\kappa , \mu )$-finitary if and only if it is both $(\kappa , \lambda )$-finitary and $(\lambda ,\mu )$-finitary.

Proof. Without loss of generality we may assume $\kappa \neq \lambda $ (otherwise, there is nothing to prove). The implication $(1) \Rightarrow (2)$ follows from Remark 9.1.9.18 (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). $\square$