Corollary 9.4.6.23. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of $\infty $-categories. Assume that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is idempotent complete. Then $U$ is an isofibration. In particular, $U$ is exponentiable.
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Proof. Using Proposition 9.4.6.22, we can factor $U$ as a composition $\operatorname{\mathcal{E}}\xrightarrow {F} \operatorname{\mathcal{E}}' \xrightarrow {U'} \operatorname{\mathcal{C}}$, where $F$ is an equivalence of $\infty $-categories and $U'$ is an isofibration. For each object $C \in \operatorname{\mathcal{C}}$, Lemma 9.4.6.20 guarantees that the induced map $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ is a Morita equivalence; in particular, every object of $\operatorname{\mathcal{E}}'_{C}$ is a retract of $F(X)$, for some object $X \in \operatorname{\mathcal{E}}_{C}$ (see Proposition 9.4.5.8). Since $\operatorname{\mathcal{E}}_{C}$ is idempotent complete, it follows that $F_{C}$ is an equivalence of $\infty $-categories. Applying Corollary 5.1.7.10, we conclude that $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$, so that $U$ is also an isofibration (Proposition 5.1.7.14). $\square$