Lifting along projective resolutions

Lemma 6.6.14 (Lifting along a projective resolution). Let \(q\colon Q_{\bullet }\to C_{\bullet }\) be a degreewise epimorphic quasi-isomorphism between bounded below chain complexes in an abelian category, and let \(P_{\bullet }\) be a bounded below projective chain complex. Every chain map \(f\colon P_{\bullet }\to C_{\bullet }\) admits a lift \(\widetilde f\colon P_{\bullet }\to Q_{\bullet }\) satisfying \(q\widetilde f=f\). Any two such lifts are chain homotopic through a chain homotopy \(h\) satisfying \(qh=0\).