$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.4.6.9. Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets which is essentially $\kappa $-small, and suppose we are given a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}& } \]
which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$. Then $U$ is flat if and only if $\widehat{U}$ is a cartesian fibration.
Proof of Proposition 9.4.6.9.
Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially $\kappa $-small inner fibration and let
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}& } \]
be a diagram which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$. It follows from Corollary 5.1.5.11 that $\widehat{U}$ is a cartesian fibration if and only if, for every $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$, the projection map $\widehat{U}_{\sigma }: \Delta ^2 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}} \rightarrow \Delta ^2$ is a cartesian fibration. By virtue of Lemma 9.4.6.13, this is equivalent to the requirement that $U$ is flat.
$\square$