# Kerodon

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Remark 8.5.5.2. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\widehat{\operatorname{\mathcal{C}}}$ be an idempotent completion of $\operatorname{\mathcal{C}}$. Then any $\infty$-category $\widehat{\operatorname{\mathcal{C}}}'$ which is equivalent to $\widehat{\operatorname{\mathcal{C}}}$ is also an idempotent completion of $\operatorname{\mathcal{C}}$. More precisely, if $G: \widehat{\operatorname{\mathcal{C}}} \rightarrow \widehat{\operatorname{\mathcal{C}}}'$ is an equivalence of $\infty$-categories, then a functor $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an idempotent completion of $\operatorname{\mathcal{C}}$ if and only if the composite functor $(G \circ H): \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}'$ exhibits $\widehat{\operatorname{\mathcal{C}}}'$ as an idempotent completion of $\operatorname{\mathcal{C}}$.