The Dual and the Double of a Hopf Algebroid are Hopf Algebroids

Let H be a x-bialgebra in the sense of Takeuchi. We show that if H is x-Hopf, and if H fulfills the finiteness condition necessary to define its skew dual H (a), then the coopposite of the latter is x-Hopf as well. If in addition the coopposite x-bialgebra of H is x-Hopf, then the coopposite of the Drinfeld double of H is x-Hopf, as is the Drinfeld double itself, under an additional finiteness condition.