Consider two cochain DGA (differential graded algebras) named $A$ and $B$. By "coproduct" of two DGA I mean the category theory coproduct, not the coalgebra coproduct. It is defined in ["A closed model structure for differential graded algebras" by J.F.Jardine][1]. 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 the coproduct of two cohomology algebras. The question now becomes whether the following statement is true

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

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

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


  [1]: https://ncatlab.org/nlab/files/JardineModelDG.pdf