I am reading Ravenel's green book(Complex Cobordism and Stable Homotopy Groups of Spheres), there is an example in its 306 page:
Let $(A, \Gamma) := (\pi_* BP, BP_* BP) \cong (\mathbb{Z}_{(p)}[v_1, v_2,...], A[t_1,t_2, ...])$, $\Sigma := A[t_{n+1}, t_{n+2}, ...] = \Gamma / (t_1, ... , t_n)$. The evident map $(A, \Gamma) \to (A, \Sigma)$ is normal.
$D := A \square_{\Sigma} A = \mathbb{Z}_{(p)}[v_1, ..., v_n]$ and $\Phi := A \square_{\Sigma} \Gamma \square_{\Sigma} A = D[t_1, ..., t_n]$.
I am trying to verify these relations about $D$ and $\Phi$.
I know the right $\Sigma$-comodule structure of $A$ and $\Gamma$ come from $\eta_R : A \to \Gamma \to \Sigma$ and $\Gamma \xrightarrow{\Delta} \Gamma \otimes_A \Gamma \to \Gamma \otimes_A \Sigma$. (left comodule structure is similar) Where $\Delta : \Gamma \to \Gamma \otimes_A \Gamma$ is the coproduct of Hopf algebroid $(A,\Gamma)$.
The theorem 4.1.18 in the book shows how these map is determined, but I don't know how to write $\eta_R$ and $\Delta$ in terms of $v_i$s and $t_i$s.