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 ≌ D ⋊ 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 M induced from C, and D ⋊ H the smash coproduct introduced by Bulacu and Nauwelaerts.