The Stacks project

Definition 12.3.9. Let $\mathcal{A}$ be a preadditive category. Let $f : x \to y$ be a morphism.

1. A kernel of $f$ is a morphism $i : z \to x$ such that (a) $f \circ i = 0$ and (b) for any $i' : z' \to x$ such that $f \circ i' = 0$ there exists a unique morphism $g : z' \to z$ such that $i' = i \circ g$.

2. If the kernel of $f$ exists, then we denote this $\mathop{\mathrm{Ker}}(f) \to x$.

3. A cokernel of $f$ is a morphism $p : y \to z$ such that (a) $p \circ f = 0$ and (b) for any $p' : y \to z'$ such that $p' \circ f = 0$ there exists a unique morphism $g : z \to z'$ such that $p' = g \circ p$.

4. If a cokernel of $f$ exists we denote this $y \to \mathop{\mathrm{Coker}}(f)$.

5. If a kernel of $f$ exists, then a coimage of $f$ is a cokernel for the morphism $\mathop{\mathrm{Ker}}(f) \to x$.

6. If a kernel and coimage exist then we denote this $x \to \mathop{\mathrm{Coim}}(f)$.

7. If a cokernel of $f$ exists, then the image of $f$ is a kernel of the morphism $y \to \mathop{\mathrm{Coker}}(f)$.

8. If a cokernel and image of $f$ exist then we denote this $\mathop{\mathrm{Im}}(f) \to y$.