?? idea of trying to turn [relationship between [?? "accidental topos" with its extra "tensor product" ...] and [ag theory of toric variety, with its extra "toric convolution" ...] .... with latter as k-module objects in former and former as special cocommutative comonoids in latter ... ??? ] into some sort of nice [adjunction and/or interpretation of doctrines ... ??? ....] ?? ....

?? comonoid process as wanting to be right adjoint here and k-module process as wanting to be left adjoint ???? .....

?? maybe implicit compatibiity conditions here ??? .... ?? between toric convolution and "tensor product" (... ??? ...) so that "tensor product" survives to comonoids .... ?? and / or between ??? and ??? so that ??? survives to ????? ...... ????? ....

?? "between cartesian product and "tensor product" so that "tensor product" survives to k-module objects" ???

??? concept of "k-module object" as involving cartesian product .... ?????? ......

?? "generalized day convolution" .... ???? .....

??? distributivities here ????? ...... ??? decategorified case ??? ... ??? pointwise product and convolution .... ??? .....