A decomposition Theorem for pointed braided Hopf algebras

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A known fundamental Theorem for braided pointed Hopf algebras states that for each coideal subalgebra, that fulfills 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 and many interesting coideal subalgebras are in particular comodules. We extend the mentioned Theorem by proving that the decomposition is compatible with this comodule structure if the underlying ordinary Hopf algebra is cosemisimple.