Twisted Hopf module coalgebras

For k a field, H a k-bialgebra and C a right H-module k-coalgebra, we define a new comultiplication on the H-module C to obtain a "twisted coalgebra" C-r. tau epsilon Hom (C, H circle times C). If tau is invertible, the categories of relative right Hopf modules over C and C-r are isomorphic. We show that crossed coproducts are invertible twistings of the tensor coalgebras. These results are dual to [1, Theroem1.1] and [1, Theorems3.4].