Kerodon

Remark 8.5.2.10. Let $\operatorname{Ret}$ denote the category introduced in Construction 8.5.0.2. Then $\operatorname{Idem}$ can be identified with the full subcategory of $\operatorname{Ret}$ spanned by the object $\widetilde{X}$. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{F}: \operatorname{Ret}\rightarrow \operatorname{\mathcal{C}}$ be the functor determined by a retraction diagram in $\operatorname{\mathcal{C}}$ (see Exercise 8.5.0.3). Then the restriction $F = \overline{F}|_{ \operatorname{Idem}}$ corresponds (under the identification of Remark 8.5.2.8) to the idempotent endomorphism of Example 8.5.2.3.