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How can I prove the following statement?

Let $\beta: B\rightarrow C$ be a quasi-isomorphism of complexes of $R$-modules. If $P$ is a complex of projective $R$-modules which is bounded below, then $\mathrm{Hom}_R(P,\beta)$ is a quasi-isomorphism; it is surjective when $\beta$ is surjective.

Any help?

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