Suppose we have modules $M_{i,j}$ over a commutative ring $R$ (or members of some abelian category, like quasi-coherent sheaves of modules), and suppose that we have a 3x3 commutative diagram, where every row and column is a short exact sequence (ommitting the extremal zeros for simplicity).
$\require{AMScd}$ \begin{CD} M_{0,0} @>>> M_{1,0} @>>> M_{2,0}\\ @VVV @VVV @VVV\\ M_{0,1} @>>> M_{1,1} @>>> M_{2,1}\\ @VVV @VVV @VVV\\ M_{0,2} @>>> M_{1,2} @>>> M_{2,2}\\ \end{CD}
By composing arrows, we obtain the (not necessarily exact) diagonal sequence \begin{equation}\label{1} 0 \to M_{0,0} \xrightarrow{\delta_0} M_{1,1} \xrightarrow{\delta_1} M_{2,2} \to 0\tag{$\Delta$} \end{equation}
Question: What is the homology of the diagonal sequence $(\Delta)$ at the location $M_{1,1}$?
It is easy to see that $\delta_0$ is injective, $\delta_1$ is surjective, and $\delta_1\circ\delta_0 = 0$. So it remains to calculate the homology of the middle term.
I am sure that there is a relevant spectral sequence or some other tool from homological algebra, but I don't know how to approach this question.