Remark 9.2.5.16 (Exactness and Flatness). Let $\kappa $ be a regular cardinal and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Using the classification of left fibrations (see Corollary 5.6.0.6), we see that the following conditions are equivalent:
- $(a)$
The functor $F$ is $\kappa $-left exact (Definition 9.2.5.10).
- $(b)$
For every uncountable regular cardinal $\lambda $ and every $\kappa $-flat functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ (Definition 9.2.4.7), the composite functor
\[ \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\xrightarrow {G} \operatorname{\mathcal{S}}_{< \lambda } \]is also $\kappa $-flat.
Moreover, if the $\infty $-category $\operatorname{\mathcal{D}}$ is locally $\lambda _0$-small (for some uncountable regular cardinal $\lambda _0$), then it suffices to verify condition $(b)$ in the special case where $\lambda = \lambda _0$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ is a corepresentable functor (Theorem 9.2.5.15).