has a standard decomposition
: ,called the basic decomposition (here , , and are respectively the image, the coimage and the reduced part of the morphism ).

If a morphism in a pre-abelian category has a nodal decomposition, then there exist morphisms and which (being not necessarily isomorphisms) connect the nodal decomposition with the basic decomposition by the following identities:
:
:
:

Categories with nodal decomposition

A category is

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