?? accidental geometric morphism ... ?? both left and right adjoint parts as preserving products, so k-module objects get preserved ... ???? so then does this give an adjunction at the k-module level ??? .... ??? and does the left adjoint preserve toric convolution ??? ... ??? .... ?????... .... ???? ....

?? look at example of affine line vs projective line ... map to terminal toric variety ... ??? ... ?? issue of preservation of affineness by imverse image ... ??? ...

?? tensoring by "toric" line bundle as preserving toric convolution ... ?? is that precisely equivalent to being cocommutative monoid ??? .... ??? .... ???