For cohomology with coefficients in a field $F$ the map $H^\cdot(X;F) \otimes H^\cdot(Y;F) \to H^\cdot(X \times Y;F)$ of the Kunneth theorem is an isomorphism of algebras over $F$. I am correct in thinking that this together with the fact that cohomology with a contravariant functor, implies that the cohomology of a topological operad is a cooperad, the dual of a notion of a cooperad?

If this is not true, where does my reasoning fail? Does it at least hold on the level of vectorspaces over $F$? If this is true, why aren't there many references about this construction? It seems that the additional algebra structure would give you some additional information.

Furthermore, using the Thom isomorphism for the cohomology, would this also imply that there is a notion of string cohomology dual to string topology?