my state of confusion at the moment is such that i've got lots of questions but have to struggle to intelligibly articulate them ... anyway i'm going to take a stab at articulating one here ....

given a bistable bimonoidal object y in a stable monoidal 2-category x, it seems reasonable (to me at the moment) to talk about y having an extra "cartesian" structure, amounting to an adjunction p between the nullary multiplication and comultiplication operations on y, together with another adjunction q between the binary multiplication and comultiplication operations on y, together with axioms saying that "for each n, there's essentially just one adjunction between the n-ary multiplication and comultiplication operations on y built from p and q".

(the case where x = the stable monoidal 2-category of categories, with cartesian product of categories as the tensor product, is supposed to explain why this terminology is reasonable; my intent is that in this case a bistable bimonoidal object y amounts to essentially just a symmetric monoidal category (with the diagonal y -> yXy as the comultiplication of the bimonoidal object), and a cartesian structure on y amounts to the property of y being cartesian (that is, that the unit object 1 in y is terminal, and that the putative projections defined using the terminalness of 1 in fact form product cones).)

whereas the 2-cells in the "walking bistable bimonoid" stable monoidal 2-category are supposed to be the isomorphisms between parallel finite spans between finite sets, the 2-cells in the "walking cartesian bistable bimonoid" stable monoidal 2-category are supposed to be the general morphisms between them, instead of just the isomorphisms. (i'm being a bit sloppy here about "cartesian vs co-cartesian" and similar distinctions, but hopefully i can get away with it for the moment without causing excessive confusion.)

(it seems to be turning out that cartesianness of bistable bimonoids is relevant to toric geometry in a somewhat different way than i'd been suspecting earlier, but hopefully i can ignore that for now ...)

ok, so if the set-up so far isn't too hopelessly screwed up yet, then i can try to ask my question here: suppose that we take x to be the stable monoidal 2-category where an object is a cocomplete category and a morphism is a cocontinuous functor and the tensor product of objects is "the walking multi-cocontinuous functor out of those objects"; then what does a cartesian bistable bimonoidal object y in x amount to? is it essentially just a cocomplete category with finite cartesian products, with cartesian product being multi-cocontinuous?