?? in trying to clarify various alleged analogies between various "combined doctrine" situations ... ?? idea that "cartesian product" recurrently shows up as "the extra product" ??? ... ?? in particular in disguised / mutated form in at least one case ??? .... ?? in 4-product case of "ag theory with extra toric convolution product", the 4 products are "tensor product", "toric convolution", "cartesian product", and "sum" ... ??? and not only is cartesian product "extra" here, but the other "extra", namely toric convolution, is itself a sort of disguised / mutated version of cartesian product ... ?? via relationship to accidental topos .... ??? ....

?? "extraness" and "flatness" here, vs "extraness" and "toricness" ??? ....

?? property/structure ambiguity (kock-zoeberlein ...) of cartesian products here (?? ...) and ... ?? "flatness" .... geometric interpretation .... ??? .....

?? hypothetical adjunction ... ?? between "_lax interchange bi-tensor category_" and "_bi-cartesian bi-tensor category_" ... ??? .... ????? ..... ???? ..... relationship of unit and/or co-unit for such to "_the_ analogy" here ??? .... "toric convolution"and cartesian product in accidental topos .... ???? .....

?? vague memory of stuff baez talked / asked about once ... ?? embellishment on right adjoint property of "cocomm comonoid(_)" ?? .... ???? bi-stable bi-monoid .... ??? ...

?? was thinking a bit about "preserving extras" in 4-product situation ... ??? but maybe not as much leeway as might guess at first .... ?? "abelian" overlap between product and sum .... ??? also using binary products as stand-in for limits with more general-shaped diagram schemes ... ??? .... ??? ....