Corollary 9.4.6.19. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. Then $U$ is flat if and only if, for every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the projection map $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \Delta ^ n$ is exponentiable. Moreover, it suffices to verify this condition in the case $n = 2$.
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