Let H be a quasi-Hopf algebra. We show that any H-bimodule coalgebra C for which there exists an H-bimodule coalgebra morphism n : C -> H is isomorphic to what we will call a smash product coalgebra. To this end, we use an explicit monoidal equivalence between the category of two-sided two-cosided Hopf modules over H and the category of left Yetter-Drinfeld modules over H. This categorical method allows also to reobtain the structure theorem for a quasi-Hopf (bi)comodule algebra given by Panaite and Van Oystaeyen, and by Dello et al.
Keywords: monoidal equivalence, (bi)comodule algebra, bimodule coalgebra, structure theorem
2010 MSC: 16W30; 18D10; 16S34
Theory and Applications of Categories, Vol. 32, 2017, No. 1, pp 1-30.