# Kerodon

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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 $\widetilde{e} \circ \widetilde{e} = \widetilde{e}$.