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
\begin{equation}
\rho\_{X\otimes Y}:=(\rho\_X\otimes\rho\_Y)\circ(1\otimes b\_{A,X}\otimes 1)\circ(\Delta\otimes 1\otimes1).
\end{equation}
The problem is that I can not quite show that it is an action. Perhaps it is only true for symmetric monoidal categories?



Do the modules over a Hopf algebra in a braided monoidal category form a monoidal category?
Yes, this is true in a braided monoidal category and your candidate for the action is correct.


A reference is Proposition 2.5 in 


**Majid, Shahn.** Algebras and Hopf algebras in braided categories. *Advances in Hopf algebras* (Chicago, IL, 1992), 55--105, Lecture Notes in Pure and Appl. Math., **158**, Dekker, New York, 1994.


