I found in the Fomenko's book that the stable cohomology operation is an inverse limit of $(H^{n+q}(K(G,n);H),f_n)$, where $G,H$ are a group (or rings, or fields for simplicity, it doesn't matter). My questions is two.
First of all, I have some problems with the definition of suspension map (for example) $$S:H^n(X;G)\longrightarrow H^{n+1}(SX;G).$$ Referring to homotopy theory I remember that when we have a map $X\longrightarrow Y$, it induces a map $SX\longrightarrow SY$, but if we use the Brown representation theorem, a map $$e \in [X,K(G,n)]= H^n(X;G)$$ by suspension is has to be a map $$Se \in [SX,SK(G,n)] \not= H^{n+1}(SX;G),$$ because $SK(G,n) \not= K(G,n+1).$
Another possibility is, by left adjoint of $S$ $$e \in [X,K(G,n)]=[X,\Omega K(G,n+1)]=[SX,K(G,n+1)] $$ goes in $Se \in [SX,K(G,n+1)]$, but in this case, this is an isomorphism at every time.
Secondly, I don't understand how the maps $f_n$ are defined. At page 394 of the Fomenko's book, he write (using the isomorphism for the cohomology operations) that $f_n^*$ is a composition of $$i_n^*:H^{n+q}(K(G,n);H) \longrightarrow H^{n+q}(SK(G,n-1);H)$$ and $$S^{-1}:H^{n+q}(SK(G,n-1);H)\longrightarrow H^{n+q-1}(SK(G,n-1);H),$$
where $i_n^*$ is determined by the condition $i_n^*F_G=SF_G $. Now, I don't understand the condition on $i_n^*$ and what it means with $S^{-1}$. I think that this second-part-problems derived by the first.
Thank you in advance.
Edit: to conclude, to prove the isomorphism between the Steenrod algebra and the cohomology (between n and 2n ) of $K(G,n)$, I have to verify that the map $f_n^*$ is the map induced by the Freudenthal map; but I think that if I understand the previously two this is simple.
I believe that the map $S$ should be the isomorphism given by your second "possibility": given $e: X \to K(G,n)$, $Se$ is the corresponding map $SX \to K(G, n+1)$. Given this, $S^{-1}$ is the inverse of this isomorphism.
The map $i_n^*$ comes from adjointness: the identity map on $K(G,n-1)$ corresponds to a map $i_n \in [SK(G, n-1), K(G,n)]$ via the bijection $$ [SK(G, n-1), K(G,n)] \leftrightarrow [K(G, n-1), \Omega K(G,n)] \leftrightarrow [K(G,n-1), K(G, n-1)]. $$ Now apply $[-, K(H, n+q)]$ to the map $i_n$ to get $$ i_n^*: [K(G, n), K(H, n+q)] \to [SK(G,n-1), K(H, n+q)], $$ or equivalently, $$ i_n^*: H^{n+q}(K(G, n), H) \to H^{n+q}(SK(G,n-1), H). $$