Variant 7.2.7.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:
- $(1)$
The functor $F$ preserves small filtered colimits.
- $(2)$
For every small filtered category $\operatorname{\mathcal{K}}$, the functor $F$ preserves $\operatorname{N}_{\bullet }(\operatorname{\mathcal{K}})$-indexed colimits.
- $(3)$
For every directed partially ordered set $(A, \leq )$, the functor $F$ preserves $\operatorname{N}_{\bullet }(A)$-indexed colimits.