Naturality of a Kunneth formula for cohomology Let $X,Y$ be CW complexes. By Kunneth formula, we have a group isomorphim
$$ H^n(X\times Y;G) \cong \oplus_{p+q=n} H^p(X;H^q(Y;G))$$
Is there a natural map realizing this isomorphism?
 A: I came across this question in my 1961 DPhil Thesis;   this was written up in two papers which are available from my Publication List, 
[3]. ``Cohomology with chains as coefficients'', Proc. London Math. Soc. (3) 14 (1964), 545-565. 
[4]. ``On K\"{u}nneth suspensions'', Proc. Camb. Phil. Soc. 60 (1964) 713-720.  
It is  shown in the first paper that the isomorphism could be chosen to be natural with respect to maps of $X$ but not with respect to maps of either $Y$ or of $G$. The naturality was important for the second paper, which was poorly titled: it should have been something like "$k$-invariants of function spaces". Paper [3] also contains as an Appendix formulae for this isomorphism in some special cases
The aim of this work was to investigate the Postnikov system of $Y^X$ by induction on the Postnikov system of $Y$. Michael Barratt suggested this as another way of determining in some cases the extensions  involved in his work on exact sequences of track groups;  such a determination in his paper "Track Groups II"  used Whitney tube systems! 
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