Lemma 12.4.2. Let $\mathcal{C}$ be a preadditive category. The following are equivalent

$\mathcal{C}$ is Karoubian,

every idempotent endomorphism of an object of $\mathcal{C}$ has a cokernel, and

given an idempotent endomorphism $p : z \to z$ of $\mathcal{C}$ there exists a direct sum decomposition $z = x \oplus y$ such that $p$ corresponds to the projection onto $y$.

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