TITLE: Relationship between universal coefficient theorem and $[K(\mathbb{Z},n), K(G,n)]$?
QUESTION [11 upvotes]: 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 $H^n(X; G)$ we get when we're changing our coefficients from $\mathbb{Z}$ to $G$ (i.e., those that come simply from tensoring with the new group) corresponded to elements of $[K(\mathbb{Z}, n), K(G, n)]$. Then, the other classes that arise from $\operatorname{Ext} /\operatorname{Tor}$ would correspond to elements of $[X, K(G,n)]$ which don't factor through $K(\mathbb{Z},n)$. Is anything like this even remotely true?
This question is in part motivated by the responses to an earlier question of mine, which mentioned that viewing $H^n(X; G)$ 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...

REPLY [2 votes]: Just a minor addition to the other answers, since you seem particularly interested in cohomology theories and operations on them.  We often concentrate on operations of a particular cohomology theory, but the theory works equally well for operations between cohomology theories.  What you describe is a particular instance of that.  Where the cohomology theories are particularly well-behaved, all this operational structure is a sort-of souped-up version of rings and modules.  You can think of the operations of a single cohomology theory as like a ring, and operations between cohomology theories as like a bimodule.  (This is an analogy, they aren't rings and bimodules, they're Tall-Wraith monoids and bimodules for such.)