Consider the following diagram which lives in the category of $R$-modules.

$$ \begin{array}{ccccccccc} 0 & \xrightarrow{i} & A & \xrightarrow{f} & B & \xrightarrow{q} & C & \xrightarrow{d} & 0 \newline & & \downarrow & & \downarrow & & \downarrow & & \newline 0 & \xrightarrow[j]{} & A & \xrightarrow[g]{} & E & \xrightarrow[r]{} & F & \xrightarrow[e]{} & 0 \end{array} $$

Let the first down arrow be an equivalence and the other down arrows be both epimorphisms.

Can we prove that the right square is a push out diagram? If not what can be said about the kernels of these two maps? Are they isomorphic?

What can be said about the diagram which is mirror revers of this diagram?