??? so to what extent have we been clear up to now that .... ?? whereas "ordinary tensor product" is given by extra left adjoint ("left-left") of essential geometric morphism, toric convolution is given by non-extra left adjoint of (in general ... in particular in non-affine case ... ?? ...) non-essential geometric morphism .... ???? ....

?? of course seems pretty obvious in various ways, but that doesn't guarantee we actually noticed it ... ??? ....

?? relationship to ... "index-raising/-lowering" issues here, and ... ??? associated relationship among various "compatibiltiy conditions" ... "lax interchange" .... ??? .....

?? can we get away here with .... talking about "toric convolution" as though it lived in topos ??? ..... ?? which maybe it sort of does, and is essentially just cartesian product ??? ..... ???? ...