Derived functors computed on projective resolutions

Exercise 6.6.18. Let \(P_{\bullet } \to C_{\bullet }\) be a quasi-isomorphism in \(\Ch (\Aa )_{\geq 0}\), with \(P_{\bullet }\) projective. Show that \(\bL F(C_{\bullet }) \simeq F(P_{\bullet })\) in \(\D ^-(\Bb )\). If \(F\) is right exact, conclude that \(\pi _0 \bL F(A) \cong F(A)\) for all \(A \in \Aa \).