Corollary 9.2.4.19. Let $\kappa $ be a regular cardinal and let $\mathscr {F}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ be a $\kappa $-flat functor, where $\lambda $ is an uncountable cardinal. Then $\mathscr {F}$ preserves all $\kappa $-small limits which exist in $\operatorname{\mathcal{D}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Enlarging $\lambda $ if necessary, we may assume that $\lambda $ is regular, that $\operatorname{\mathcal{D}}$ is essentially $\lambda $-small, and that $\lambda $ has exponential cofinality $\geq \kappa $ (so that the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$ admits $\kappa $-small limits). Using Lemma 9.2.4.17, we can realize $\mathscr {F}$ as the colimit of a $\lambda $-small $\kappa $-filtered diagram
\[ \operatorname{\mathcal{K}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}_{< \lambda } ) \quad \quad \alpha \mapsto \mathscr {F}_{\alpha }, \]
where each $\mathscr {F}_{\alpha }$ is corepresentable by an object of $\operatorname{\mathcal{D}}$ and therefore preserves all limits which exist in $\operatorname{\mathcal{D}}$ (Proposition 7.4.1.18). Since $\operatorname{\mathcal{K}}$-indexed colimits commute with $\kappa $-small limits in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$ (Proposition 9.1.5.8), it follows that the functor $\mathscr {F}$ preserves $\kappa $-small limits. $\square$