?? another piece of propaganda for idea that semi-monoidalness of accidental topos isn't that fundamental .... ??? that the _essentialness_ of the "binary multiplication" geometric morphism is crucial in getting tensor product of toric quasicoherent sheaves to exist ... ??? .....

?? we sort of did implicitly almost notice this before ... ??? topos whose diagonal is non-essential, but with essential co-diagonal (though peculiarly not in co-nullary case ... "co-nullary co-diagonal" doesn't exist as geometric morphism ... ???? ....) .... ???? .....

???? so _does_ "second right adjoint of tensor product" generally exist in algebraic geometry ?? ... construing tensor product as working on bimodules .... ?? so i guess that i'm really asking about first right adjoint of pullback along diagonal ...... ????

?? some confusion here ... ???

??? toric case ... length 3 adjoint string ... middle (?? in affine case ... ??? ....) = "pull back along mult hom mXm->m to turn m-set into [mXm]-set .... ??? .....
left-left adjoint as .... ??? ....

??? same triple reverse confusion we keep running into ??????? ......

?? topos-wise we have "tensoring of torsors" as an essential geometric morphism ....
?? the left-left adjoint is "tensor product of filteredly-cocontionuous set-valued functors .... the left-right adjoint is ... "pull back along tensor product of torsors to turn fccsvf of one variable into fccsvf of two variables ... ???

?? toric-wise we have .... left-left adjoint and left-right adjoint together form toric geometric map ....

topos-wise we have f_!, f^*, f_* ??? ....

toric-wise we have f^*, f_*, f^! ??? .....



??? how do general "toric maps" (?? ...) get along with toric convolution, and to what extent does this explain (?? or make even more confusing .... ???? ....) relationship / overlap between toric maps and topos-theoretic geometric morphisms ??? .....

???? functors that get along (?? ...) with toric convolution but _not_ especially with ordinary tensor product .... ?? relationship to topos-theoretic geometric morphisms that .... ?? well, that stretch the relationbship to "toric maps" .... ??? ....

??? hmmm .... ??? so "topos-wise f^*, f_* lining up with toric-wise f^*, f_*" as "combined doctrine" idea ... ?? comparatively rare but "nice" ... ??? ... ?? and fits with (or maybe _is_ ... ???? .... coincides with ... ??? ....) "single functor preserving both ordinary tensor product and toric convolution" .... ????? ....

while, "topos-wise f_!, f^*, f_* lining up with toric-wise f^*, f_*, f^!" is more like "combined mayhem" ... ?? more frequent and "normal", but .... ?? often confusing for perhaps obvious reasons .... ???? ...

?? and then there's the idea that maybe, at least in the toric case, "everything factors into combined mayhem followed by combined doctrine" ...... ????? ..... ??? understanding better why this happens, or at least whether it really does .... ??? .... ?? also understanding overlap / residue given by "line bundles" or whatever ... ??? ....

(?? relationship to general topos pattern of surjection-injection image factorization ???? .... ?? ... adjunction ... co-/monad .... ??? ....)

??? then ... ??? questions about possible non-toric analogs of everything (?? ...) here .... ????? .....

?? in particular, some question that i may have been trying to get at in the first place here .... ?? as to geometric interpretation of "f^!" in general algebraic geometry .... ???? any relationship to "inverse image preserves affineness" ??? ..... ???? ..... and extent to which tendency for ag geometric diagonal to have f^! holds .... ??? and relationship to stuff that simon what's-their-name may have tried to explain to us .... ??? "grothendieck's six operations" ?? ... ??? ... ??? .....