Let $\mathcal{A}$ be an abelian category (hence every object in $\mathcal{A}$ has a projective resolution), let $M$ and $N$ be two objects in $\mathcal{A}$. Consider the following commutative diagram where each row is a projective resolution and the existence of $f_0, f_1, \dots$ is given by comparison theorem.

Do we have the following or similar conclusion?

The map $f$ is a monomorphism if and only if the following sequence is exact: $$ \cdots \xrightarrow{} P_1\oplus Q_2 \xrightarrow{\left[ \begin{smallmatrix} \epsilon _1& 0\\ -f_1& \eta _2\\ \end{smallmatrix} \right] } P_0\oplus Q_1 \xrightarrow{\left[ \begin{smallmatrix} f_0& \eta_1 \end{smallmatrix} \right] } Q_0. $$

And how to prove that?

Enkidu told me I should consider Horseshoe lemma, but I don't know what to do.

Thank you very much.