Corollary 9.4.6.28. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of simplicial sets and suppose we are given a digram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{\widehat{U}} \\ & \operatorname{\mathcal{C}}& } \]
which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise idempotent completion of $\operatorname{\mathcal{E}}$. Then $\widehat{U}$ is a flat isofibration.