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 this paper by J.F.Jardine. And cohomology of an algebra $A$ is defined by $H(A)=ker(d)^{i+1}/Im(d)^{i}$. 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

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

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.