Four-Lemma of modules of monomorphisms
Proposition
Let 
be an abelian category and the following commutative diagram in
where 
are monomorphisms and 
is an epimorphisms.
Suppose the rows are exact, then 
is a monomorphism
Proof
w.l.o.g. for 
, the general case follows from Mitchell's embedding theorem.
Let 
.
Then it remains to show, that 
.
Then by commutavity, we get
and since 
is a monomorphism, we get
or 
.
Thus by exactness, there exists an 
with 
.
Suppose w.l.o.g. that 
, otherwise 
Then by injectivity, 
with
and thus 
.
Hence by exactness, there exists an 
with 
and by surjectivity of there exists an 
with 
.
Note that
and thus 
.
Furthermore, since exactness implies
it follows, that
