For any complex of $R$-modules, $P$ and $M$, $\hom_R(P,M)$ is the complex defined by,
$$\forall n \in \mathbb{Z}\ \ \ \hom_R(P,M)_n = \prod_{i \in \mathbb{Z}} \hom_R(P_i, M_{i+n})$$

We say that a complex $P$ is $\pi$-projective, if for all quasi-isomorphisme $f: M \to M'$ 
$$\hom_R(P,f): \hom_R(P,M') \to \hom_R(P,M)$$
is a quasi-isomorphisme to.

Equivalently we says that the complex of $R$-modules $P$ is $\pi$-projective, if for all exacte complex $M$, $\hom_R(P,M)$ is also exacte.

Now suppose $f:P \to P'$ is a quasi-isomorphism of $\pi$-projective complexes of $R$-modules. I want to show that for any complex of $R$-module M, $\hom_R(f,M)$, which is at degrée $n$:
\begin{align}
\hom_R(f, M)_n :\hom_R(P', M)_n &\longmapsto \hom_R(P, M)_n\\
(\alpha_i : P'_i \to M_{i+n})_{i \in \mathbb{Z}} &\longmapsto (\alpha_i \circ f_i )_{i \in \mathbb{Z}}
\end{align}
is a quasi-isomorphism.