Structure Theorem for Dual Quasi-Hopf Bicomodules and Its Application

We present a structure theorem for dual quasi-Hopf bicomodules, and also obtain the structure theorem $C\cong D\rtimes H$ for dual quasi-Hopf module coalgebras, where $H$ is a dual quasi-Hopf algebra, $C$ a right $H$-module coalgebra, and $D$ a left $H$-comodule coalgebra in the tensor category $^{H}\mathcal{M}$ induced from $C$, and $D\rtimes H$ the smash coproduct introduced by Bulacu and Nauwelaerts.