?? so .... ?? topos picture equivalent of "ag morphism whose right adjoint is strong monoidal wrt toric convolution" as .... ??? strong monoidal left adjoint between accidental toposes, whose right adjoint ... ??? well, maybe two possibilities here, and maybe i think that i know which one is correct ... not the one that i was originally guessing .... vacuous property (my original guess ... "strong monoidalness of arbitrary right adjoint wrt cartesian product as automatic" ... ??? .... but cartesian product vs its cocontinuous extension (again ...) ... ??? this sort of thing as complicating things here again .... ???? ....

some of the ideas sloppily stated here ... as usual ... uncertainties all around ... ??? ....

?? so when does "homming from a bi-presheaf" (in one variable ...) get along nicely (?? ...) with cocontinuous extension of cartesian product ?? .... ?? just that diagonal / diagonal pun ???? .... ??? .... hmm, how messed up _is_ that pun ?? ....

?? vacillating over vacuousness here ... ?? possibility that some tameness factor is making something vacuous here even though the vacuousness arguments that we've been trying to give don't quite make sense ??? ... ??? .... ??? ....


?? toric convolution vs cocontinuous extension thereof ... ??? ....

?? cocontinuous extension of cartesian product (?? along universal bicocontinuous ?? ...) in "discrete" case as "taking diagonal of matrix of sets" .... ????? ....

?? "discrete" case as somewhat antithetical to case we might be interested in ??? ...

?? arbitrary strong symmetric monoidal functor between affine accidental toposes as coming from comm monoid hom ??? ... ?? because of left universal property of accidental topos .... categorified monoid alg of single object smc .... .... ??? ....

?? "fourier duality" argument for why "toric ag morphism" should have right adjoint strongly monoidal wrt toric convolution ?? ... ??? ....