I'm currently educating myself about homological algebra from Weibel's book (motivated by an interest in Floer and Khovanov knot homologies). After reading about long exact sequences in homology I became curious about what the boundary maps can tell us about the morphisms in the sequence (assuming the objects and maybe one of the maps is known).
More formally, let $\mathcal{A}$ be an abelian category, and let $C(\mathcal{A})$ be its complex category. We know that if $0\to A\to B\to C\to 0$ is a short exact sequence in $C(\mathcal{A})$ then there are boundary morphisms $\delta_n$ such that $$\ldots\to H_n(A)\to H_n(B) \to H_n(C) \overset{\delta_n}{\to} H_{n-1}(A)\to\ldots$$ is a long exact sequence in $\mathcal{A}$.
Assume I am given two short exact sequences $0\to A\overset{f}{\to} B\overset{g}{\to} C\to 0$ and $0\to A\overset{f'}{\to} B\overset{g'}{\to} C\to 0$ (same objects, possibly different morphisms) whose boundary morphisms turn out to be the same.
Does this tell me anything about the relation between $f$ and $f'$ or $g$ and $g'$? What if I assume that $f=f'$ or $g=g'$?
