I've started studying (stable) cohomological operations on my lecture notes, and I was given that an equivalent definition for a family of cohomological operations to be a stable cohomological operation was to satisfy the following relation: $$\Sigma (\tau_n (\imath_{A,n}))=\rho_{n+1}^*(\tau_{n+1}(\imath_{A,n+1}))$$ where the notation is defined ad follows: $\Sigma \colon \tilde{H}^n(K(A,n))\to \tilde{H}^{n+1}(\Sigma K(A,n))$ is the suspension isomorphism.
$K(A,n)$ is the Eilenberg-Maclane space of type $A,n$, which comes together with a preferred isomorphism $\phi_{n}\colon \pi_n(K(A,n),*)\to A$ (I just fix it once and for all).
$\imath_{A,n}$ is the fundamental class in $H^n(K(A,n),n)$ which is the representing object for the functor $H^n(-,A)$, and can be defined as $\Phi^{-1}(\phi_n \circ H^{-1})$,where $\Phi$ is the morphism given by UCT and $H$ is Hurewicz (from homotopy to homology).
$\rho \colon \Sigma K(A,n) \to K(A,n+1)$ is defined via the adjunction $$ \hom (\Sigma K(A,n), K(A,n+1)) \cong \hom(K(A,n), \Omega K(A,n+1))$$ as the map adjoint to $\hat{\rho} \colon K(A,n) \to \Omega K(A,n+1)$ which is uniquely defined by its action on the $n$-th homotopy groups $$ \pi_n(K(A,n),*) \xrightarrow{\phi_{n}} A \xrightarrow{\phi_{n+1}^{-1}} \pi_{n+1}(K(A,n+1),*) \cong \pi_n(\Omega K(A,n+1),*)$$
Back to the claim: The proof of the above equality relies on the following fact which I'm unable to prove (once one has proved it, it's easy to prove the characterisation) $$\rho^*(\imath_{A,n+1})=\Sigma \imath_{A,n}$$
MY ATTEMPT: what I can do is some juggling with naturality and boil everything down to prove $$ \Sigma \Phi^{-1}(\phi_n \circ H^{-1})= \Phi^{-1}(\phi_{n+1}\circ \rho_{H}\circ H^{-1})$$ where $\rho_H$ is the map induced in Homotopy. But now I can't do anything, I mean, for sure I'm lacking of some properties but I can't find them anywhere. I'm unable to find a reference on where this result is given. I think the key point is that I don't know how to control $\rho_H$, because it's defined as the adjoint of a map over which I've full control on what it does on homotopy, but I don't know how to transfer this knowledge via the adjunction.
ADDENDUM: My definition of stable cohomological operation of degree $n$ is a family of reduced cohomological operations $\{\tau_i\}_{i}$ of degree $n$ between $\tilde{H}^i(-,A) \to \tilde{H}^{i+n}(-,B)$ which commutes with the suspension isomorphism.
