Given two cochain DGA (differential graded algebra) named $A$, $B$. For coproduct of two DGA I mean category theory coproduct, not coalgebra's coproduct. It is defined in "A closed model structure for differential graded algebras" by J.F.Jardine. And cohomology of an algebra $A$ is defined by $H^{i + 1}(A)=\ker(d^{i+1})/\operatorname{Im}(d^{i})$. It is a natural DGA with zero differential. Then we can define coproduct of two cohomology algebra. The question now becomes: Whether the following statement is true

$$H(A) \amalg H(B) \cong H(A \amalg B)$$

where isomorphism is taken under $\mathbf{DGA}_k$.

I think it is true at least for DG algebra with underlying structure free algebra, with zero differential. I have written a proof. But I am still checking it.