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 $K$ of $f$ is the equalizer of $f$ and the zero morphism $0: A \to B$ together with the morphism $K \to A$ coming from the definition of equalizer.

On Uniqueness
Since the kernel is defined by a universal property it is only unique up to unique isomorphism.

While for example in group theory the kernel of a group homomorphism $f : G \to H$ is a subset of $G$, not all categorical kernels of $f$ in the category of groups are subsets of $G$.

Also see

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