Abstract
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 is equivalent to the existence of a suitable map S: H → H that we call a preantipode of H.
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