Assume $(H,m,\eta, \Delta, \epsilon, S,r)$ to be a coquasitriangular Hopf-Algebra over $\mathbb C$. The category $C(H)$ of $H$-comodules is braided monoidal.
Now consider a coaction $\delta: H \rightarrow H \otimes_{\mathbb C} H$ of $H$ on itself such that $H$ is a comodule-algebra (algebra object in $C(H)$ . Can the antipode $S$ and comultiplication $\Delta$ of $H$ be "braided" in such a way that $\underline H:=(H, m, \eta, \underline \Delta, \epsilon, \underline S)$ in $C(H)$ will be a Hopf-Algebra object in $C(H)$?
To be a bit more precise I am asking for a way to define a new comultiplication $\underline \Delta$ and antipode $\underline S$ using the given braiding and existing comultiplication and antipode such that $\underline H$ is a Hopf algebra in $C(H)$.
This should be "similar" to transmutations of hopf algebras