??? so ... vague memory of paper giving "humble adjunction ... homming of just plain categories ..." as extreme special case of "structure / semantics adjunction" ... (?? maybe gray or someone ??? ....) ... ?? i guess now that that was all about "single-universe semantics" ... formula category vs model category ....

_cat_^op -> _cat_

x |-> _set_^x

?? "flip level slips" here ??? .... ??? ....

?? anyway, "homming into k (?? mainly k=_set_ ??) as contravariantly self-adjoint" ??? ....

?? then .... require formula category be c-complete for some class c of diagram schemes .... ???? .... ?? what should happen to model category then ???? ..... ????? ....

?? not really sure i'm that close yet to what they wrote in that paper .... ??? ....

?? maybe try working out "products" case to get more of an idea .... ??? ....

?? _should_ colimits show up here too ?? ... ??? ... i meant in formula cat, along with limits, but then also, model cat .... ???? ...

?? multi-universe semantics .... ??? "naturality" .... ???? ....

??? "underlying set of model" .... ???? .... "concrete operation" .... ??? "naturality" .... ???? ..... "structure theory of object in functor category" ..... ?????? ..... ???? .... monad .... ???? ..... ?? where monad wants to "be interpreted" / "algebras" ..... ???? .....

?? "threshold where model cat and formula cat look same" .... ??? .... ???? .... ??? "contravariance" here ?? ... ??? .....