Corollary 9.3.4.23. Let $\kappa $ be a regular cardinal and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ 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 $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small, and that $\lambda $ has exponential cofinality $\geq \kappa $ (so that the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$ is $\kappa $-complete). Using Lemma 9.3.4.21, we can realize $\mathscr {F}$ as the colimit of a $\lambda $-small $\kappa $-filtered diagram
\[ \operatorname{\mathcal{K}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \lambda } ) \quad \quad \alpha \mapsto \mathscr {F}_{\alpha }, \]
where each $\mathscr {F}_{\alpha }$ is corepresentable by an object of $\operatorname{\mathcal{C}}$ and therefore preserves all limits which exist in $\operatorname{\mathcal{C}}$ (Proposition 7.4.1.22). Since $\operatorname{\mathcal{K}}$-indexed colimits commute with $\kappa $-small limits in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$ (Proposition 9.1.5.10), it follows that the functor $\mathscr {F}$ preserves $\kappa $-small limits. $\square$