Given two monoidal categories $\mathcal{M}$ and $\mathcal{N}$, can one form their tensor product in a canonical way? 


The motivation I am thinking of is two categories that are representation categories of two algebras $R$ and $S$, and the module category of the tensor product algebra $R \otimes\_{\mathbb{C}} S$.


Also, if $\mathcal{M}$ and $\mathcal{N}$ are assumed to be braided monoidal, can we tensor the braidings?



The tensor product of two monoidal categories
The book [Tensor Categories](http://www-math.mit.edu/~etingof/egnobookfinal.pdf) discusses, with many variations, the details of Robert McRae's answer. Just like for vector spaces, there are a number of related but inequivalent "tensor products" of linear categories, with the choice dependent on the types of linear categories considered: sufficiently finite dimensional (vector spaces / categories) have only one reasonable tensor product, but the "algebraic" tensor product of infinite-dimensional objects can be completed in various ways. Locally finite abelian categories is a particularly good choice.


Once you have made such a choice, tensoring (braided) monoidal structures is typically easy. You probably will need to require that the monoidal structure is "continuous" for however you chose to complete your tensor products. Again, see the Tensor Categories book for details.


