Hello

I've done an exercise in which the 3rd question was the lemma of 5 itself. Now, I've done it in a non conventional way and my teacher says it's wrong but she didn't point out exactly what was wrong with my solutions. I'll just post one of them and please tell me where exactly it fails, if it indeed fails (I'm still convinced it doesn't fail):

Hhmm. Imagine the

from

to

. Now, the exercise is: Consider the above commutative diagram in which each horizontal sequence is exact. If

is an epimorphism and

is a monomorphism, then

.

My proposal of solution doesn't need that

is a monomorphism, and is done in the following way:

Where is this wrong, if at all?