Preantipodes for Dual Quasi-Bialgebras

It is known that a dual quasi-bialgebra with antipode H, i.e. a dual quasi-Hopf algebra, fulfils a fundamental theorem for right dual quasi-Hopf H-bicomodules. The converse in general is not true. We prove that, for a dual quasi-bialgebra H, the structure theorem amounts to the existence of a suitable map S : H --> H that we call a preantipode of H