Construction 8.5.2.7 (The Universal Idempotent). We define a category $\operatorname{Idem}$ as follows:
The category $\operatorname{Idem}$ has single object $\widetilde{X}$.
Morphisms in $\operatorname{Idem}$ are given by $\operatorname{Hom}_{\operatorname{Idem}}( \widetilde{X}, \widetilde{X}) = \{ \operatorname{id}_{\widetilde{X}}, \widetilde{e} \} $.
The composition law on $\operatorname{Idem}$ is given (on non-identity morphisms) by $\widetilde{e} \circ \widetilde{e} = \widetilde{e}$.