Why is the Steenrod algebra isomorphic to the cohomology of the Eilenberg MacLane spaces?

Question

I will stick to $p=2$.

I define the Steenrod algebra to be the algebra of (topological) stable cohomology operations modulo 2.

I've found in the literature the identification of the Steenrod algebra $\mathcal{A}^*$ with the cohomology (of a spectrum) $H\mathbb{F}_2^*(H\mathbb{F}_2)$. In particular, this means that $\mathcal{A}^n$ is identified with $\pi_{-n} \textrm{map} (H\mathbb{F}_p, H \mathbb{F}_p)$.

Could someone explain this identification? Moreover, is there some good reference that proves it?

Answer 1

This is an application of Yoneda's lemma: if a functor is represented by an object $H$, then the set of natural transformations from that functor to itself is given by $\mathrm{Hom}(H,H)$. In this case, the stable cohomology operations are precisely the natural transformations from $H\mathbb{F}_2^*$ to itself.