TITLE: Tensor product of abelian categories
QUESTION [10 upvotes]: It seems the notion of tensor product of abelian categories exists naturally. 
Does someone know the reference of the construction?

REPLY [3 votes]: Yes Julian you are exactly right. In the paper you cite the Deligne product is extended to general (bi-)module categories in a way that reduces to the product of abelian categories in the case that the monoidal category involved is $Vec$. There it is also shown that certain classical formulas (adjunction of hom and tensor, for example) hold on the level of module categories. 
In an upcoming article the tensor product of module categories is studied from the perspective that it is a special case of a generalization of the center of a module category as introduced by Drinfeld and studied to great effect by Müger et. al., the so called relative center.