?? sym mon cocontinuous fr from [g,_set_] to _set_ .... ???? and / or to accidental topos (with tensor product ...) .... ??? cocomm comonoid adjunction bit here .... ???? ....

?? left-universal property of [g,_set_] as sym mon cocomplete cat .... ??? relationship to "double coset" / "orientation" logic .... ???? .... ?? "torsor" .... ???? ..... ?? "g-action x st x#x = g-fold sum of x, via certain cocone ..." .... ????? ....

?? adjunction confusion again .... ???? ..... ?? to accidental topos, vs to underlying sym mon cocomplete cat of ag th ... ?? possibly one coming from a tag theory ??? ..... ??? .... ??? revisiting more than one old confusion here ?? ... ?? relationship between them ?? ..... ????? ....... ?? "richer doctrine / poorer environment" (?? ...) confusion .... ??? vs .... ???? .... toric stack confusion ..... ????? ......

?? different concepts of "same theory but in different doctrine" ... ??? .... decategorified analog .... ???? ....... underlying cat / set changes, vs structure (...) on it changes .... ???? .... ???? ..... ????? sounds a bit like theory vs environment ?????? .... ????? .......



??? "toric rigidity" .... ?????? .....