Zoran Skoda comonad for comodule algebra

Let an algebra $E$ has two $H$-comodule structures, $\rho_1,\rho_2$ with isomorphism of corresponding comonads $\alpha: G_1\to G_2$. This induces an isomorphism of $k$-modules $\alpha_E:E\otimes H\to E\otimes H$ underlying $G_1(E)\to G_2(E)$, and let $\alpha_E(1\otimes 1) = z$. Then for all $e\in E$, taking into account that $\alpha_E$ is a morphism of $E$-modules,

$$\alpha_E(\rho_1(e)) = \alpha_E(e\triangleright_1 (1\otimes 1)) = e\triangleright_2 \alpha_E(1\otimes 1) = \rho_2(e) z$$

As the natural transformation has to commute with the counit of the comonad, we read in particular that $(id_M \otimes \epsilon) \alpha_M = id_M \otimes \epsilon$ on $M\otimes H$ for each $E$-module $M$. In particular, $z-1 \in Ker(id_E \otimes \epsilon)$. On the other hand, $\alpha_E$ has to correctly commute with $id \otimes \Delta$, what imposes that if $z = \sum z^1 \otimes z^2$, then $1 \otimes 1 \mapsto (\alpha_E\otimes id_H)(1\otimes \Delta(1)) = \sum z^1 \otimes z^2\otimes 1$, but also $1\otimes 1\mapsto (id_E\otimes\Delta)\alpha_E(1\otimes 1) = (id_E \otimes \Delta)(z) = \sum z^1 \otimes z^2_{(1)}\otimes z^2_{(2)}$. Act with $\id \otimes \epsilon\otimes \id$ to obtain $(\sum z^1 \epsilon(z^2)) \otimes 1 = \sum z_1 \otimes z_2 = z$. Thus $z = z' \otimes 1$ what with $z-1 \in Ker(id_E \otimes \epsilon)$ implies $z' = 1$; hence also $z = 1$, therefore $\alpha_E(\rho_1(e)) = \rho_2(e)$.