Proposition 3

Let $f : M' \to M$ be a homomorphism of A-modules, $p' : P' \to M'$ a morphism into $M'$ of a complex null on the right and projective $P'$, and $p : P \to M$ a left resolution of M. There exists a morphism of complexes $\tilde{f} : P' \to P$, and only one up to homotopy, such that $p \circ \tilde{f} = f \circ p'$.

Let us consider the complex $\overline{P}$ defined as follows: $\overline{P}_n = P_n$ for $n \neq -1$, $\overline{P}_{-1} = M$, $d_{\overline{P},n} = d_{P,n}$ for $n \neq 0, -1$, $d_{\overline{P},0} = p_0$, $d_{\overline{P},-1} = 0$, and the complex $\overline{P}'$ defined analogously. Applying to the complexes $\overline{P}$ and $\overline{P}'$ Prop. 1 a) with $r = 0$, $u_i = 0$ for $i < -1$ and $u_{-1} = f$, we obtain Prop. 3.