A coequalizer of a pair of maps 
in a category is a
map 
such that
1. 
, where 
denotes composition.
2. For any other map 
with the same property, there is exactly one map 
such that 
i.e., one has the above commutative
diagram.
It can be shown that the coequalizer is an epimorphism and that, moreover, it is unique up to isomorphism.
In the category of sets, the coequalizer is given by the quotient set
 |
and by the canonical map 
, where 
is the minimal equivalence relation on 
that identifies 
and 
for all 
.
The same construction is valid in the categories of additive groups, modules, and vector
spaces. In these cases, the cokernel of a morphism 
can be viewed, in a more abstract categorical
setting, as the coequalizer 
of 
and the zero map.
The dual notion of the coequalizer is the equalizer.
