Editing Cokernel (section)

=== Special cases ===
In a [[preadditive category]], it makes sense to add and subtract morphisms. In such a category, the [[coequalizer]] of two morphisms {{mvar|f}} and {{mvar|g}} (if it exists) is just the cokernel of their difference:

: <math>\operatorname{coeq}(f, g) = \operatorname{coker}(g - f).</math>

In an [[abelian category]] (a special kind of preadditive category) the [[image (category theory)|image]] and [[coimage]] of a morphism {{mvar|f}} are given by
:<math>\begin{align}
    \operatorname{im}(f) &= \ker(\operatorname{coker} f), \\
  \operatorname{coim}(f) &= \operatorname{coker}(\ker f).
\end{align}</math>

In particular, every abelian category is normal (and conormal as well). That is, every [[monomorphism]] {{mvar|m}} can be written as the kernel of some morphism. Specifically, {{mvar|m}} is the kernel of its own cokernel:
:<math>m = \ker(\operatorname{coker}(m))</math>