Corollary 7.1.4.15. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an isofibration of $\infty $-categories. Then $U$ is a coreflective localization functor if and only if, for every object $Y \in \operatorname{\mathcal{D}}$, the fiber $\operatorname{\mathcal{C}}_{Y} = \{ Y\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ contains a $U$-initial object of $\operatorname{\mathcal{C}}$.
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Proof. Assume that $U$ is a coreflective localization functor. We will show that, for each object $Y \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{Y}$ contains a $U$-initial object of $\operatorname{\mathcal{C}}$ (the converse follows immediately from Proposition 7.1.4.13). Using Proposition 7.1.4.13, we see that there exists a $U$-initial object $X \in \operatorname{\mathcal{C}}$ and an isomorphism $e: Y \rightarrow U(X)$ in $\operatorname{\mathcal{D}}$. Since $U$ is an isofibration, we can lift $e$ to an isomorphism $\widetilde{e}: \widetilde{Y} \rightarrow X$ in the $\infty $-category $\operatorname{\mathcal{C}}$. Our assumption that $X$ is $U$-initial then guarantees that $\widetilde{Y}$ is also $U$-initial (Corollary 7.1.4.11). $\square$