??? generalizing from flat co-presheaf (??by "co-presheaf" here i really mean ... ???that it could be valued in any co-complete environment ... ???? ?? well, perhaps before the generalization i assume the environment is actually a topos, but not afterwards ... ???? ... ..... ??? vs ... ???actually using galois connection involving just actual "classical-valued" co-presheaves .... ???? ....) to non-flat, vs generalizing from monic co-yoneda morphism to non-monic ... ???might part of the point be that to get non-trivially different galois connection you need to make both of these generalizations ??? ... ???? ....

???try examples ... ????....


(??? any sort of "ternary galois connection" (???) going on here??? ... formula, model, co-yoneda morphism .... ???? ....

??? ?? "gabriel-ulmer triality" ???? .... ??? ..... ?? though ... ?? extent to which (??some?? ...) stuff that we're trying to work out here is covered under "gabriel-ulmer duality" heading as ... ???maybe affected by ... "size issues" ... ???? ..... ??? .... ????? ??? 2-topos ??? .... "doctrine" .... ??? ... (??? idea of trying to _define_ doctrine as special kind of 2-topos ??? ..... ???? and doctrine morphism as .... ???? special kind of geometric morphism ??? ... ???? .... ??extent to which this is already part of what lurie (or maybe others) worked out ... ?? maybe for somewhat similar reasons to why i'm trying to work it out .... ??? one reason for wondering about this being ... ??possibility of cheap way of getting help with working out some "size" issues ??? ... ??? .....)

??? truth-value-valued pairing between model and co-yoneda morphism ... ??
??? set-valued pairing between model and formula ... ??
??? "codomain" projection from co-yoneda morphisms to formulas ... ????

??? actually, probably still fair amount of sloppiness / confusion here from "grothendieck vs lawvere-tierney" issues .... ???

?? "candidate for closed" ... ????formula ???

?? "candidate for dense" ... ??? or "candidate for dense in given candidate for closed" ... ??? "co-yoneda morphism into formula" ???? ....


)

(??generalizing from co-yoneda morphism to arbitrary morphism between presheaves ... ??? .... ???grothendieck vs lawvere-tierney ?? ... ??? "auxiliary types" ???? .... ?? ... hmmm ... sensible morphism from one formal colimit to another ... ??? if we argue that this can be reduced to co-yoneda morphism (?? using "auxiliary types" trick or "lawvere-tierney" trick ... ???? ...), then why not argue further that it could be reduced down to just ordinary morphism ??? ...... ???? .... ??? hmm, or is there something in the obviously "asymmetric" nature of colimits that actually makes it sensible that ... ?? you should only attempt the one of the two "symmetric" sorts of reduction here ?? ... ???? .....)