In short, I'm wondering whether the universal coefficient theorem can be understood/reinterpreted by using maps of Eilenberg-MacLane spaces.  This is a wishy-washy idea and I don't have evidence to back it up, but it would be very nice if the "freebie" cohomology classes in H<sup>n</sup>(X;G) we get when we're changing our coefficients from &#8484; to G (i.e., those that come simply from tensoring with the new group) corresponded to elements of [K(&#8484;,n),K(G,n)].  Then, the other classes that arise from Ext/Tor would correspond to elements of [X,K(G,n)] which <i>don't</i> factor through K(&#8484;,n).  Is anything like this even remotely true?

This question is in part motivated by the responses to [an earlier question of mine][1], which mentioned that viewing H<sup>n</sup>(X;G) as [X,K(G,n)] helps us understand cohomology operations (in that case, Steenrod squaring).  It seems as if the representability of cohomology is probably only useful for studying honest cohomology operations, but I don't think I understand exactly what that means well enough to deduce whether changing coefficients qualifies...

  [1]: https://mathoverflow.net/questions/461/understanding-steenrod-squares