# Do the modules over a Hopf algebra in a braided monoidal category form a monoidal category?

Given a (strict) braided monoidal category $(\mathcal{C},\otimes,I)$ with braiding $b$ and a Hopf algebra $H$ in $\mathcal{C}$. There is a category Rep($H$) of modules over $H$ in $\mathcal{C}$. Do these have a canonical monoidal structure? My candidate for an action on $X\otimes Y$ for $H$-modules $X$ and $Y$ is $$\rho_{X\otimes Y}:=(\rho_X\otimes\rho_Y)\circ(1\otimes b_{A,X}\otimes 1)\circ(\Delta\otimes 1\otimes1).$$ The problem is that I can not quite show that it is an action. Perhaps it is only true for symmetric monoidal categories?