Definition:Abelian Category

Definition 1
An abelian category is a pre-abelian category in which:
 * every monomorphism is a kernel
 * every epimorphism is a cokernel

Definition 2
An abelian category is a pre-abelian category in which:
 * every monomorphism is the kernel of its cokernel
 * every epimorphism is the cokernel of its kernel

Definition 3
An abelian category is a pre-abelian category in which for every morphism $f$, the canonical morphism from its coimage to its image $\operatorname{coim}(f) \to \operatorname{im}(f)$ is an isomorphism.

Also see

 * Equivalence of Definitions of Abelian Category