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Proposition Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the core $\operatorname{\mathcal{C}}^{\simeq }$ is a replete subcategory of $\operatorname{\mathcal{C}}$ (Example that is, the inclusion $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ is an isofibration of $\infty $-categories

Proof. Combining Example, Remark, and Remark, we deduce that the inclusion map $\operatorname{\mathcal{C}}^{\simeq } \hookrightarrow \operatorname{\mathcal{C}}$ is an inner fibration; in particular, $\operatorname{\mathcal{C}}^{\simeq }$ is an $\infty $-category. The repleteness is immediate from the definition (since $\operatorname{\mathcal{C}}^{\simeq }$ contains every isomorphism in $\operatorname{\mathcal{C}}$). $\square$