Corollary 9.4.6.14. Every functor of $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ is a flat inner fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Choose an uncountable regular cardinal $\kappa $ for which $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Using Theorem 9.4.4.1, we can choose a diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \Delta ^1 & } \]
which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$. It follows from Proposition 9.4.1.11 that $\widehat{U}$ is a cartesian fibration, so that $U$ is flat by virtue of Proposition 9.4.6.9. $\square$