For $U_q(\frak{g})$ the Drinfeld--Jimbo quantum group, its category of representations is equivalent to the category of representations of $U(\frak{g})$, or equivalently the category of Lie algebra representations of $\frak{g}$. Both categories have an obvious monoidal structure, what is not obvious is if this is an equivalence of monoidal categories.

Edit: As Phil's comment below, this is not a monodial equivalence. As the linked answer says, the problem is that the associators in both cases are different. What is the easiest way to see that this is so? The discussion about 6j symbols and coordinates on stacks is unfortunately lost on me. In fact, for modules over a Hopf algebra the assoicators look easy to the naive untrained eye, why is it about the assoiators on either category that is non-trivial?