Lemma 9.3.4.21. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\lambda $-small and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Then a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ is $\lambda $-flat if and only if there exists a $\lambda $-small $\kappa $-filtered diagram $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ such that $\mathscr {F}$ is the colimit of $(h_{\bullet } \circ U): \operatorname{\mathcal{E}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda })$.
Proof. The “if” direction follows by combining Lemma 9.3.4.20 with Example 9.3.4.11. For the converse, we proceed as in the proof of Corollary 9.3.4.15. Assume that $\mathscr {F}$ is $\kappa $-flat, so that the $\infty $-category
is $\kappa $-filtered (Remark 9.3.4.13). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be the projection map. Since the covariant Yoneda embedding $h_{\bullet }$ is dense (Variant 8.4.2.4), $\mathscr {F}$ is a colimit of the diagram $h_{\bullet } \circ U$. To complete the proof, it will suffice to show that the $\infty $-category $\operatorname{\mathcal{E}}$ is essentially $\lambda $-small. This follows from Corollary 4.9.8.12, since $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small and the right fibration $U$ is essentially $\lambda $-small. $\square$