The cokernel of a group homomorphism 
of Abelian groups
is the quotient group 
. For a module
homomorphism of modules (or a linear
transformation of abstract vector spaces),
the same construction gives a quotient module
(or quotient vector space, respectively).
More generally, let 
be a morphism in a category
with zero morphisms. These are distinguished morphisms 
whose composition
on either side with any compatible morphism is again
zero. A cokernel of 
is a morphism 
such that 
and every morphism 
satisfying 
factors uniquely as
for a morphism 
. This universal
property determines the cokernel up to a unique isomorphism. In particular, it applies in a preadditive
category without requiring a zero object (Stacks
Project Authors 2026).
