Definition:Kernel (Category Theory)/Definition 2

Definition
Let $\mathbf C$ be a category.

Let $A$ and $B$ be objects of $\mathbf C$.

Let $f: A \to B $ be a morphism in $\mathbf C$. Let $\mathbf C$ have a zero object $0$.

A kernel of $f$ is a morphism $\ker(f) \to A$, which is an equalizer of $f$ and the zero morphism $0: A \to B$.

Also see

 * Equivalence of Definitions of Kernel of Morphism
 * Definition:Cokernel (Category Theory)
 * Definition:Image (Category Theory)
 * Definition:Coimage (Category Theory)