Understanding proof of Universal coefficient theorem for cohomology I am working through Cohomology chapter on Hatcher's book and I am having trouble with the proof of Universal Coefficient theorem for Cohomology. To be concrete I don't understand the last part of the proof of the Lema 3.1 (b). It is the highlighted  text on the picture below:

My problems arise since I don't see why having the identity $\alpha \colon H \to H'$ can extend to maps between all the groups (modules) between the resolutions such that the induced maps are isomorphisms in cohomology. (I hope I have managed to explain my problem). 
Thanks in advance. 
Note:
I'm proving it for modules over PIDs but that shouldn't change anything. 

This is the fundamental lemma in homological algebra (see here theorem $2.22$ page $36$ and the following corollary.)
If you take the identity the thesis of the theorem says in particular that you obtain a chain homotopy
