Let $H$ be a bialgebra over $\mathbb{C}$. Suppose that $V$ is a left $H$-comodule and $W$ is an $H$-module. Then we can defined map $\Psi$ by \begin{align} & \Psi: V \otimes W \to W \otimes V, \\ & \Psi(v \otimes w) = v_{(-1)}.w \otimes v_{(0)}. \end{align}
Suppose that the map $\Psi$ is given and the map $\delta: V \to H \otimes V$, $\delta(v) = v_{(-1)} \otimes v_{(0)}$, is given. Can the formula $\Psi(v \otimes w) = v_{(-1)}.w \otimes v_{(0)}$ determine the action $H \otimes W \to W$?
It seems that the formula $\Psi(v \otimes w) = v_{(-1)}.w \otimes v_{(0)}$ determines the action $H \otimes W \to W$. Since $\Psi$ and $\delta$ are linear maps, we can choose a basis of $V$ and a basis of $W$ and write down the map $\Psi$ explicitly and we obtain a system of equations. Then we can solve the system of equations and obtain the action $H \otimes V \to V$. Is this true? Thank you very much.
Edit: let $H=\mathbb{C}[GL_n]$ and $W = V$ the vector representation of $GL_n$ and $\delta(v_i) = \sum_{j} c_{ij} \otimes v_j$ ($v_1, \ldots, v_n$ is a basis of $V$). In this case, can the formula $\Psi(v \otimes w) = v_{(-1)}.w \otimes v_{(0)}$ determine the action $H \otimes W \to W$?
I think that in this case we have \begin{align} & \Psi( v_i \otimes v_j ) \\ & = (v_i)_{(-1)}.v_j \otimes (v_i)_{(0)} \\ & = \sum_l c_{il}.v_j \otimes v_l \\ & = \sum_l \sum_k p_k^{(ijl)} v_k \otimes v_l, \end{align} for some $p_k^{(ijl)}$.
On the other hand, since $\Psi$ is given, we have \begin{align} \Psi(v_i \otimes v_j) = \sum_{k,l} q_{kl}^{(ij)} v_k \otimes v_l, \end{align} for some $q_{kl}^{(ij)}$. Therefore we have $$ p_k^{(ijl)} = q_{kl}^{(ij)}, \quad i,j,k,l = 1, \ldots, n. $$ The action of $H$ is given by $c_{il}.v_j = \sum_k p_k^{(ijl)} v_k$.
