Remark 9.2.4.12. Example 9.2.4.11 admits a partial converse. Let $\lambda $ be an uncountable regular cardinal and let $\operatorname{\mathcal{D}}$ be an idempotent complete $\infty $-category $\infty $-category which is essentially $\lambda $-small. For any functor $\mathscr {F}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}_{< \lambda }$, the $\infty $-category of elements $\int _{\operatorname{\mathcal{D}}} \mathscr {F}$ is also idempotent complete (Corollary 8.5.4.24) and essentially $\lambda $-small (Corollary 4.7.9.12). It follows that $\int _{\operatorname{\mathcal{D}}} \mathscr {F}$ is $\lambda $-cofiltered if and only if it has an initial object (Proposition 9.1.8.10). In other words, $\mathscr {F}$ $\lambda $-flat if and only if it is corepresentable.
More generally, if we assume that $\operatorname{\mathcal{D}}$ is essentially $\lambda $-small (but not necessarily idempotent complete), then a functor $\mathscr {F}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ is $\lambda $-flat if and only if it is a retract of a corepresentable functor. This follows from Theorem 9.2.4.14 (applied in the special case $\kappa = \lambda $), together with Example 9.2.1.10.