Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.
This article concerns itself with the properties that hold specifically because of the existence of kernels and cokernels.
(The alternative term "difference kernel" for binary equalisers derives from this fact.)
Since pre-abelian categories have all finite products and coproducts (the biproducts) and all binary equalisers and coequalisers (as just described), then by a general theorem of category theory, they have all finite limits and colimits.
The existence of both kernels and cokernels gives a notion of image and coimage.
Note that this notion of image may not correspond to the usual notion of image, or range, of a function, even assuming that the morphisms in the category are functions.
For example, in the category of topological abelian groups, the image of a morphism actually corresponds to the inclusion of the closure of the range of the function.
In many common situations, such as the category of sets, where images and coimages exist, their objects are isomorphic.
Recall that all finite limits and colimits exist in a pre-abelian category.
In general category theory, a functor is called left exact if it preserves all finite limits and right exact if it preserves all finite colimits.
In a pre-abelian category, exact functors can be described in particularly simple terms.
Then it turns out that a functor between pre-abelian categories is left exact if and only if it is additive and preserves all kernels, and it's right exact if and only if it's additive and preserves all cokernels.
is quasi-abelian if and only if all kernel-cokernel pairs form an exact structure.
An example for which this is not the case is the category of (Hausdorff) bornological spaces.
[2] The result is also valid for additive categories that are not pre-abelian but Karoubian.
Pre-abelian categories that are not abelian appear for instance in functional analysis.