Dual quasi-Hopf algebras and antipodes

Let H be a coalgebra. In this paper, we show that H is a dual quasi-bialgebra if and only if the category M-H of comodules is a tensor category; and H is a braided dual quasi-bialgebra if and only if MH is a braided tensor category. If H is a braided dual quasi-Hopf algebra. it is shown that the antipode of H is inner, i.e., s(2)(h) = Sigma tau(h(1))h(2)tau(-1)(h(3)).