Example 8.5.2.2 (Identity Morphisms). Let $\operatorname{\mathcal{C}}$ be a category. For every object $X \in \operatorname{\mathcal{C}}$, the identity morphism $\operatorname{id}_{X}: X \rightarrow X$ is an idempotent endomorphism in $\operatorname{\mathcal{C}}$. Conversely, if $e: X \rightarrow X$ is an idempotent endomorphism in $\operatorname{\mathcal{C}}$ which is also an isomorphism, then $e = \operatorname{id}_{X}$.
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