This is exercise 6.2 in Paolo Aluffi's Algebra: Chapter 0 (the second print).
Assuming that $\mathsf{A}$ has enough projectives, prove that the full subcategory $\mathsf{\hat{A}}$ of $\mathsf{K^{-}(P)}$ consisting of complexes with cohomology concentrated in degree $0$ is equivalent to $\mathsf{A}$. (Prove that $H^0$ is fully faithful on this subcategory.)
So suppose we have two complexes $P^{\bullet}$ and $Q^{\bullet}$ consisting of projective objects. To show $H^0$ is full, we need show that for any given morphism $f: H^0(P^{\bullet}) \to H^0(Q^{\bullet})$, there exist a morphism $F:P^{\bullet} \to Q^{\bullet}$ inducing $f$.
Now I can prove this in the case that $P^i=Q^i=0$ for $i>0$, using lemma 6.3. (In this case, both complexes are actually resolutions of their $0$-th cohomology.)
But I can't see how to prove this if $P^{\bullet}$ and $Q^{\bullet}$ are just complexes with cohomology concentrated in degree $0$ (this is more general than the above case). Could you help me? Thanks in advance.