Proposition 9.1.9.10. Let $\kappa $ and $\lambda $ be regular cardinals satisfying $\kappa \triangleleft \lambda $. For every $\infty $-category $\operatorname{\mathcal{C}}$, the following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\lambda $-small $\kappa $-filtered colimits.
- $(2)$
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\operatorname{N}_{\bullet }(A)$-indexed colimits, for every $\lambda $-small $\kappa $-directed partially ordered set $(A, \leq )$.
Moreover, if these conditions are satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-finitary if and only if it preserves $\operatorname{N}_{\bullet }(A)$-indexed colimits, for every $\lambda $-small $\kappa $-directed partially ordered set $(A, \leq )$.