Let $(E,\mu,e)$ be a flat homotopy commutative ring spectrum, so we have an isomorphism
$$\Phi_E:E_*E\otimes_{\pi_*E}E_*E\to E_*(E\wedge E)$$
sending homogeneous elements $x:S^n\to E\wedge E$ and $y:S^m\to E\wedge E$ to the composition
$$S^{n+m}\xrightarrow\cong S^n\wedge S^m\xrightarrow{x\wedge y}E\wedge E\wedge E\wedge E\xrightarrow{E\wedge\mu\wedge E}E\wedge E\wedge E$$
(see Proposition 2.2 here).
Explicitly, here $E_*E$ is a right $\pi_*E$-module via the map $\eta_R:\pi_*E\to E_*E$ induced under $\pi_*$ by the monoid homomorphism
$$E\xrightarrow\cong S\wedge E\xrightarrow{e\wedge E}E\wedge E,$$
and is a left $\pi_*E$-module via the map $\eta_L:\pi_*E\to E_*E$ induced under $\pi_*$ by the monoid homomorphism
$$E\xrightarrow\cong E\wedge S\xrightarrow{E\wedge e}E\wedge E.$$
Then we may define a "comultiplication"
$$\Psi:E_*E\to E_*E\otimes_{\pi_*E}E_*E$$
as the composition
$$E_*E\xrightarrow{E_*(e\wedge E)}E_*(E\wedge E)\xrightarrow{\Phi_E^{-1}}E_*E\otimes_{\pi_*E}E_*E.$$
Then it is asserted in the literature that this map is co-associative, i.e., it makes the following diagram commute:
I am struggling to show this, and I was wondering if anyone could outline the argument or provide a reference that does more than just assert this.
The only place I can recall seeing this done carefully is in Lecture 3 of Adams's "Lectures on generalised cohomology. In Category Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three". Available here, but presumably with some institutional access needed.