I am currently learning about the theory of monoidal categories, and something is bothering me. I want to define the opposite category $C^{op}$ of a (braided) monoidal category. My guess would be that given a category, tensor product, unit, left/right unitors, associativity, and braiding $(C,\otimes,1,\lambda,\rho,\alpha,\beta)$ , one could define the opposite by $(C^{op},\otimes^{op},1^{op},\lambda^{op},\rho^{op},\alpha^{op},\beta^{op})$, where the $op$ is taken appropriately on the level of objects, categories, functors, and natural transformations.
Something about this bugs me though. On nLab's "opposite category page" (https://ncatlab.org/nlab/show/opposite+category), they say that $(C^{op})^{op}\neq C$ when $C$ is a non-symmetric braided category. I thought that the double-opposite of objects/categories/functors/natural transformations bring you back where you started. Hence, there must be something else afoot.
What is the correct definition of the opposite category of a (braided) monoidal category?
As motivation, I am looking at (braided) monoidal categories with duals. In this case, the duality extends to a functor $(-)^{*}:C^{op}\to C$. The dual also respects monoidal and braided structure in some sense, and so I would like to say that $(-)^{*}$ induces a fully faithful braided monoidal functor $C^{op}\to C$. It seems like this might even be an equivalence of categories (in the case of fusion categories I can give a proof), but I don't know for sure.