?? ag morphism whose right adjoint is strong monoidal wrt toric convolution ??? .... ?? is this a/the correct idea ??? ..... ??? ....

?? idea of "trying to squeak by with just the middle morphisms" as maybe ruined / complicated by ... ??? ... possible problematicness of "pairing" here ??? ....

?? "operations" sometimes partial ... in one (?? ...) case "cohomological" ... ?? morphisms allowed to be cohomological in general .... ???? .....

?? possible clue to "stuff needed for theory" (?? ...) in this "cohomological" bit ??? .... ???? ....

?? how "semi-monoidal topos" (...) bit compares / relates to other approaches to developing structure of toric category that we've been pursuing ... ?? ...