A decomposition Theorem for pointed braided Hopf algebras

A known fundamental Theorem for braided pointed Hopf algebras states that for each coideal subalgebra, that fulfils a few properties, there is an associated quotient coalgebra right module such that the braided Hopf algebra can be decomposed into a tensor product of these two. Often one considers braided Hopf algebras in a Yetter-Drinfeld category of an ordinary Hopf algebra. In this case the braided Hopf algebra is in particular a comodule, as well as many interesting coideal subalgebras. We extend the mentioned Theorem by proving that the decomposition is compatible with this comodule structure if the underlying ordinary Hopf algebra is cosemisimple.