Smash (co)products and skew pairings

Let $\tau$ be an invertible skew pairing on $(B,H)$, where $B$ and $H$ are Hopf algebras in a symmetric monoidal category ${\mathcal C}$ with (co)equalizers. Assume that $H$ is quasitriangular. Then we obtain a new algebra structure such that $B$ is a Hopf algebra in the braided category ${}_{H}^{H}{\mathcal Y}{\mathcal D}$ and there exists a Hopf algebra isomorphism $w\colon B\infty H \rightarrow B \bowtie_\tau H$ in ${\mathcal C}$, where $B\infty H$ is a Hopf algebra with (co)algebra structure the smash (co)product and $B \bowtie_\tau H$ is the Hopf algebra defined by Doi and Takeuchi.