Proposition 9.4.7.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^2$ be a functor of $\infty $-categories. Then $U$ is flat if and only if, for every object $X \in \operatorname{\mathcal{E}}$ satisfying $U(X) = 0$, the inclusion functor
is left cofinal.
Proof. Choose an uncountable regular cardinal $\kappa $ such that $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Using Lemmas 9.4.7.2 and 9.4.7.3, we see that $U$ is flat if and only if, for every object $X \in \operatorname{\mathcal{E}}$ satisfying $U(X) = 0$, the corepresentable functor $h_{X}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ has the property that $h_{X}|_{ \operatorname{\mathcal{E}}_{\geq 1} }$ is left Kan extended from $\operatorname{\mathcal{E}}_{1}$. Since $h_{X}$ is a covariant transport representation for the left fibration $\operatorname{\mathcal{E}}_{X/} \rightarrow \operatorname{\mathcal{E}}$, this is equivalent to the requirement that $\iota _{X}$ is left cofinal (see Corollary 7.4.3.15). $\square$