For any complex of $R$-module M, how to show that $\hom_R(f, M)$ is a quasi-isomorphism if $f:P \to P'$ is a quasi-isomorphism of $\pi$-projective complexes of $R$-modules. 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: $$ \hom_R(P,M) = \prod_{i \in \mathbb{Z}} \hom_R(P_i, M_{i+n})$$