equalizer and equal maps
1. Proposition
Let 
be a category, 
be objects, 
morphisms and 
the equalizer.
TFAE:
is an equalizer
are isomorphic- the map
is an epimorphism
2. Proof
2.1. 1) 
2)
by assumption,
commutes
Suppose there exists an object 
and a morphism 
Then there exists only one choice
making the diagram commute
2.2. 2) 3)
follows from
2.3. 3) 4)
follows from isomorphism as epimorphism
2.4. 4) 1)
By assumption, since
commutes, we conclude, that
Since 
is epi, it follows by definition, that
