Definition 8.5.2.1. Let $\operatorname{\mathcal{C}}$ be a category. An idempotent endomorphism in $\operatorname{\mathcal{C}}$ is a pair $(X,e)$, where $X$ is an object of $\operatorname{\mathcal{C}}$ and $e: X \rightarrow X$ is an endomorphism of $X$ which satisfies the identity $e = e \circ e$, so that we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{e} & \\ X \ar [ur]^{e} \ar [rr]^{e} & & X. } \]
In this situation, we will also say that $e$ is an idempotent endomorphism of $X$.