Remark 9.2.4.13. Let $\lambda $ be an uncountable regular cardinal and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ be a functor of $\infty $-categories. Then $\mathscr {F}$ can be realized as the contravariant transport representation of a right fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, which is well-defined up to equivalence (Proposition 8.6.8.3). In this case, the functor $\mathscr {F}$ is $\kappa $-flat (in the sense of Definition 9.2.4.7) if and only if the $\infty $-category $\operatorname{\mathcal{E}}$ is $\kappa $-filtered (in the sense of Variant 9.1.1.4).
In particular, if $\operatorname{\mathcal{C}}$ is locally $\lambda $-small and $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda })$ is a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$, then $\mathscr {F}$ is $\kappa $-flat if and only if the $\infty $-category $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda }) } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda })_{ / \mathscr {F} }$ is $\kappa $-filtered. See Corollary 8.4.2.7.