coequalizer in Set
1. Proposition
Given the category set, sets 
and maps 
, the coequalizer is the set
for the smallest equivalence relation such that 
with the usual projection 
as morphism 1
see: proof
2. Proof
2.1. equivalence relation
note that this is reflexive (
), transitive (existence of a zigzag) and symmetric (swapping 
)
2.2. cofork
and 
2.3. universal property
Suppose
Then the map
is the mediating morphism.
2.3.1. independent of representatives
Suppose 
, then one of the following cases applies
2.3.1.1. 1
, hence 
2.3.1.2. 2
there exists a zigzag 
, such that 
.
Thus by 
, we conclude that
2.3.1.3. 3 + 4
there exist such that 
and 
Since we get
2.3.2. unique
note that 
is forced by 
Footnotes:
1
see proof for a constructive definition not requiring Zorn's lemma