I want to construct a chain of morphisms from a dg algebra $A$ to $B$. I assume that $A$ and $B$ is non positive, i.e, $A^n$ vanishes for $n$ greater than zero. What I have is that $H^*(A)$ is isomorphic to $H^*(B)$ as a graded algebra. Here $H^*(A)$ is the graded homology algebra associated with $A$. Now the Kadeishvili Theorem show that we have $H^*(A)\rightarrow A$ and $H^*(B)\rightarrow B$ are quasi-isomorphism of $A_{\infty}-$algebras. Can I deduce that $H^*(A)\rightarrow H^*(B)$ is a morphism of $A_{\infty}-$algebras?
Added after edit: Now suppose that we have an equivalence of categories $\mathcal D(A)\rightarrow \mathcal D(B)$ which sends A to B. Then is the above question true? We know from A infinity structure on Ext algebras that the higher multiplications of $H^*(A) $ are essentially the same with the Massey products. Now the question is whether such an equivalence ‘preserves’ Massey products.