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?
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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), 55105, Lecture Notes in Pure and Appl. Math., 158, Dekker, New York, 1994.

$\begingroup$ Or look at [Majid, Shahn. Braided groups. J. Pure Appl. Algebra 86 (1993), no. 2, 187221]. Lemma 1.1. is a comodule version though $\endgroup$ – Zahlendreher Jul 20 '17 at 15:28

1$\begingroup$ Thank you a lot. This is exactly what I was looking for. $\endgroup$ – BGJ Jul 20 '17 at 18:32