Four-Lemma of modules of epimorphisms
Proposition
Let 
be an abelian category (resp. category Group) and the following commutative diagram in
where 
are epimorphisms and 
is a monomorphism.
Suppose the rows are exact, then 
is an epimorphism
Proof
w.l.o.g. we continue to assume that 
.
The general case follows from Mitchell's embedding theorem
Then by exactnesss and 
there exists an 
such that
By surjectivity of 
, there exists an 
with 
.
Then by commutativity,
hence 
Suppose 
, then 
and thus by surjectivity of 
there exists an 
such that 
Furthermore by exactness, it holds that 
and thus also by commutativity
Since 
, as is injective, we conclude that 
and therefore 
or by exactness
Thus there exists an 
such that
or by commutativity
Hence
and by choosing 
with 
we get
or 