As I am trying to get a better grasp of cohomology and exact sequence I am reading through the book mentioned in the title.
In the construction of the connecting homomorphism I understand the recipe of the zig-zag lemma. But as an exercises it is left to show that such map is well defined. And to do so we need to show that the cohomology class of $[a]$ is independent by the choice of the representative of $[c] \in H^k(\mathcal{C})$ and $b \in B^k$.
My attempt was to re-do similar calculations but in terms of cosets rather than single elements. I didn't really end up with anything, or at least I don't know how to draw the final conclusion.
Can anyone provide me hints please? (I'd like to give it a one more go).