The Structure Theorem for Weak Module Coalgebras

Let $H$ be a weak Hopf algebra, let $C$ be a weak right $H$-module coalgebra, and let $\overline C=C/C\cdot \operatorname{Ker}\operatorname{\sqcap}^{L}$. We prove a structure theorem for weak module coalgebras, namely, $C$ is isomorphic as a weak right $H$-module coalgebra to a weak smash coproduct $\overline C\times H$ defined on a $k$-space
$$ \{\Sigma c_{(0)}\otimes h_2\varepsilon(c_{(-1)}h_1)\mid c\in C,\,h\in H\} $$
if there exists a weak right $H$-module coalgebra map $\phi\colon C\to H$.