Let $A$ be a finite dimensional algebra over field K. $M,N$ are $A$-modules such that $0 \rightarrow P_1 \rightarrow P_0 \rightarrow M \rightarrow 0$ and $0 \rightarrow Q_1 \rightarrow Q_0 \rightarrow N \rightarrow 0$ are minimal projective resolutions of them, respectively. Let $f: M \rightarrow N$ be an morphism. Then by the projectivity of $P_1,P_0$, we have $f_1,f_0$ such that the following diagram commutes:
Also we know that $f_1,f_0$ may not be unique. I want to know under the condition that $Hom_A(M,Q_0)=0$, can we get $f_0,f_1$ uniquely determined by $f$?