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

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.

remark: $$ \hom_R(P,M) = \prod_{i \in \mathbb{Z}} \hom_R(P_i, M_{i+n})$$

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.