Lemma 12.3.11 (0E43)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 12: Homological Algebra
Section 12.3: Preadditive and additive categories
Lemma 12.3.11 (cite)

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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.

Proof. Part (1) follows easily from the uniqueness required in the definition of a kernel. The proof of (2) is dual. Part (3) follows from (2), since the coimage is a cokernel. Similarly, (4) follows from (1). $\square$

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