Warning 9.4.6.21. Suppose we are given a commutative diagram of $\infty $-categories
where $U$ and $U'$ are inner fibrations and $F$ is an equivalence of $\infty $-categories (but not necessarily an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$). Lemma 9.4.6.20 asserts that if $U$ is flat, then $U'$ is also flat. Beware that the converse generally does not hold: if $U'$ is flat, then $U$ need not be flat. For example, suppose that $U'$ is an isomorphism and that $F: \operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{E}}'$ is an isomorphism from $\operatorname{\mathcal{E}}$ to a full subcategory of $\operatorname{\mathcal{E}}'$. If $F$ is essentially surjective, then it is an equivalence of $\infty $-categories. In this case, the inner fibration $U = U' \circ F$ is flat if and only if $F$ is an isomorphism.