TITLE: "The Z/2-cohomology functor from Top to GrVecSpaces factors through Unstable-A-Mod, and this is the largest algebraic category through which it factors."
QUESTION [9 upvotes]: (Here A is the Steenrod algebra.)
I have it on good authority (p. 23) that this is true, but I can't quite make sense of it.  The crux of the matter is something I've been wondering for a while:

Why exactly is it that cohomology
  operations are "the best we can do"
  (i.e./more generally, why is it that
  the finest structure we can impose on
  $E$-cohomology is that it be a module
  over $E^*E$)?

There might be something tied up in the qualification that we're talking about algebraic categories, which are defined here.  I've never heard of these before.  Presumably their properties show up in GrVecSpaces as manifestations of actual things on the topological side, which makes it seem like we're only restricting ourselves to categories that are going to preserve some information that we obviously (?) want to preserve, but probably there's more to it than that.  And in any case I don't understand why those should be exactly the properties of GrVecSpaces that we care about.
(I'd love to hear about category theory stuff of course, but my main goal is to understand the boxed question.)

REPLY [4 votes]: A little more information is at the page on infinitary Lawvere theories on the nLab.  Basically, Charles is right: as cohomology is defined by the Eilenberg-Mac Lane spaces, everything about the resulting algebraic category is contained in the Eilenberg-Mac Lane spaces and their morphisms.  In particular, if that category had more structure then that would be visible in the cohomology of the Eilenberg-Mac Lane spaces and would then be already in the algebraic category we were already in.  In short it's because (as I proved on that page, in generality) the cohomology of the Eilenberg-Mac Lane spaces are the free algebras in this algebraic category.
Furthermore, to extend a point that Sam Isaacson makes, it's not true to say that $E^\star(X)$ is a module over $E^\star(E)$ with the usual interpretation of module as an abelian group with a (suitable) action of a ring.  (Charles' answer is correct in his language, but you may not have noticed his choice of language.)  What is correct is to say that $E^\star(X)$ is a module for the Tall-Wraith monoid $E^\star(\underline{E}_\star)$.  This use of module generalises the idea of a module over an algebra.  In particular, the fact that $E^\star(X)$ is an $E^\star$-algebra is included in this description.  More on the nLab page Tall-Wraith monoids and the papers linked therein.
(All of this holds for any generalised cohomology theory.)