Kerodon

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Remark 8.5.2.8. Let $\operatorname{\mathcal{C}}$ be a category and let $e: X \rightarrow X$. be an idempotent endomorphism in $\operatorname{\mathcal{C}}$. Then there is a unique functor $F: \operatorname{Idem}\rightarrow \operatorname{\mathcal{C}}$ satisfying $F(\widetilde{X}) = X$ and $F( \widetilde{e}) = e$.