Kerodon

Remark 8.5.3.2. Let $\operatorname{\mathcal{C}}$ be a category. It follows from Remark 8.5.2.8 (and Proposition 1.3.3.1) that evaluation on the morphism $\widetilde{e} \in \operatorname{Hom}_{\operatorname{Idem}}( \widetilde{X}, \widetilde{X})$ supplies a bijection from the set of idempotents in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 8.5.3.1) to the set of idempotent endomorphisms $(X,e)$ in the category $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.5.2.1). We can therefore view Definition 8.5.3.1 as a generalization of Definition 8.5.2.1.