Lemma 12.3.11. Let $\mathcal{C}$ be a preadditive category. Let $f : x \to y$ be a morphism in $\mathcal{C}$.

If a kernel of $f$ exists, then this kernel is a monomorphism.

If a cokernel of $f$ exists, then this cokernel is an epimorphism.

If a kernel and coimage of $f$ exist, then the coimage is an epimorphism.

If a cokernel and image of $f$ exist, then the image is a monomorphism.

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