Definition:Kernel (Category Theory)/Definition 2

Definition
Let $\mathbf C$ have an zero object $0$.

A kernel of $f$ is the 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)