It is well known (and wouldn't be so-named unless it were) that:

If $\xi$, $\eta$ are $n$-fold extensions of $N$ by $M$ (modules over a ring $R$) which yield the same element of $\text{Ext}^n(M,N)$, then they are in fact equivalent.

I am trying to understand a certain proof of this fact (and indeed the proof, for me, is as important as the result). My question involves a number of commutative diagrams, so I have decided to post it as a Latex pdf file. You can find it at:

http://homepages.utoledo.edu/mcrumle/mathoverflow%20letter.pdf

The proof is textbook (literally), and I am sure that I am simply having a brain fart over the thing, but it has been torturing me for some time. I'm hoping that someone can lead me out of the forest.

Thanks in advance for any help.