Corollary 9.2.5.23 (066Y)

Corollary 9.2.5.23. Let $\kappa $ and $\lambda $ be regular cardinals, where $\lambda $ is uncountable, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ be a functor of $\infty $-categories. Each of the following conditions implies the next:

$(1)$: The functor $F$ is $\kappa $-left exact.
$(2)$: The functor $F$ is $\kappa $-flat.
$(3)$: The functor $F$ preserves all $\kappa $-small limits which exist in $\operatorname{\mathcal{C}}$.

If $\operatorname{\mathcal{C}}$ admits $\kappa $-small limits, then all three conditions are equivalent.

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 9.2.5.16 and the implication $(2) \Rightarrow (3)$ is Corollary 9.2.4.19. If $\operatorname{\mathcal{C}}$ admits $\kappa $-small limits, then the implication $(3) \Rightarrow (1)$ is a special case of Theorem 9.2.5.22. $\square$