?? had vague idea that progressively higher spans might arise from / relate to simply (?? ...) "refining the adjunction between multiplication and comultiplication to a progressively higher adjoint adjunction ..." (?? ...) but .... ??? unclear ?? .... ??? as to whether the pattern involves stuff that doesn't really parse nicely for just a lone adjunction .... ???? .... ???? ....

?? cartesian bimonoid x as stable bimonoid x with stable comonoid structure on generic object x -> x ?? ... ??? .... ??? microcosmic parsing requirements here ??? ..... ??? alleged bit about ... ?? needing cartesianness of x in order to define object of natural transformations between given parallel pair of x-enriched functors ... ??? .... ?? other alleged bit about monoidalness of [comonoid y, monoid z] .... ??? hmmm, maybe relevant .... ????? .....

?? "equality becoming isomorphism" vs "morphism becoming arity" .... ??? "x-oidal d-theory" for x theory of doctrine (?? ...) d ?? ...

?? "comultiplication" adjoint (?? ...) to multiplication but without bimonoid compatibility between them .... ??? ....

?? stable bimonoid (?? in stable 2-cat ...).... ??? .... with unit and counit forming adjunction between nullary mult and comult, and another unit and counit forming adjunction between binary mult and comult .... ??? hmmm, and then maybe the axioms to say that "there's essentially just one adjunction for each arity" amount to coassociativity (?? and cocommutativity ???? .... ????) clause for certain comonoid structure .... ???? ...... ... to try to anticipate certain hopefully obvious questions i was going to ask .... ???? .... ??? relationship of all this to naive "diagonal" and "projection" .... ???? .... ??? lurking "codiagonal" and "coprojection" ????? ..... ???? .....

?? "distributive category" ... ???? as cartesian in some nice (?? maybe obvious ??) way ?? ...

????? hopf object in hopf cat vs in cartesian cat ...... ????? ...... ????? .... ???? ...... ???? bimonoid in monoidal cat vs in bimonoidal cat .... ???? .....

?? "stable monoid with stable monoid structure on generic object" ....

?? "stable monoid with stable monoid structure on generic object and stable monoid structure on generic generic object" .......

?? "stable monoid with stable monoid structure on generic object and stable monoid structure on generic generic object and ......" .......

????? ......

??? bicartesian structure on 2-category of cartesian categories ??? .... ???? .....

?? "variable abstract category" ..... ???? ....