An equalizer 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 equalizer is a monomorphism. Moreover, it is unique up to isomorphism.
In the category of sets, the equalizer is given by the set
 |
and by the inclusion map 
of the subset 
in 
.
The same construction is valid in the categories of additive groups, rings,
modules, and vector spaces.
For these, the kernel of a morphism 
can be viewed, in a more abstract categorical setting, as
the equalizer 
of 
and the zero
map.
The dual notion is the coequalizer.
