Let $$A \xrightarrow{f} B \to C $$ and $$A \xrightarrow{g} B \to C $$ be two short exact sequences of Abelian groups (or more generally, any Abelian category $\mathcal A$). I would like to ask the following question:
Is it true that this can always be completed as a commutative diagram with vertical isomorphisms? (Sorry that I didn't draw a commutative diagram here, since my familiar way - using tikzcd - seems does not work in this website...)
(Edit: as in the comment point out, I should add the finitely generated condition here)
I am thinking of this as a baby version of the following question:
Let $A \xrightarrow{f} B$ be a two-term complex in $D^b(\mathcal A)$. Is it true that the three objects $A,B$, and the mapping cone $C(f)$ together determine the isomorphism class of the two-term complex $A \xrightarrow{f} B$ in $D^b(\mathcal A)$?
I guess both questions have negative answers, but I didn't figure out an example... Thanks in advance for any (partial) answers or comments.
