Let be a monoidal category (in fact one can easily modify all statements to generalize all statements here to bicategories). One can consider several variants of the 2-category of all categories with monoidal action of , (co)lax monoidal functors and their transformations. A category with an action of is sometimes called a -actegory. The word ‘module category’ over is also used, specially when the category acted upon is in addition also additive, like the examples in [representation theory]].
If a -actegory is a categorification of a module, then for two monoidal categories and , we should categorify a bimodule, which we call --biactegory. The two actions on a usual bimodule commute; for biactegories the commuting is up to certain coherence laws, which are in fact the expression of an invertible distributive law between the two monoidal actions. The tensor product of biactegories can be defined (here the invertibility of the distributive law is needed) as a bicoequalizer of a certain diagram.
For very basic outline see section 2 in
A longer 2006 preprint called “Biactegories” (around 17 pages) has never been finished (one old version: link)
In a language of “module categories”, a different treatment is now available in