Example 9.4.6.10. Suppose we are given a commutative diagram of $\infty $-categories
where $F$ is fully faithful and induces a Morita equivalence $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$, for each object $C \in \operatorname{\mathcal{C}}$. If $\kappa $ is an uncountable regular cardinal such that $U'$ is essentially $\kappa $-small, then any fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}'$ can also be regarded as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$ (see Proposition 9.4.5.10). Applying the criterion of Proposition 9.4.6.9, we conclude that $U$ is flat if and only if $U'$ is flat. In particular, an inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is flat if and only if its fiberwise idempotent completion is flat.