so i'm getting a bit confused here ... about locally finitely presentable categories, vs their opposites ... and stuff like that ... ???...

??consider a small finite limits theory t ... and consider its models in the not-so-small finite limits environment _set_ ... ?? ...

?? but also consider the free arbitrary limits theory on t ... ????....

let's try an example; say t is the theory of categories ... ??...

??so what is "the free arbitrary limits theory on t" like?? ... assuming that it actually makes sense ... ??does it make sense??? ...

let's see, it seems plausible that taking the contravariant set-valued functors on x which take a given class of cocones to limit cones amounts to taking the universal cocompletion of x in which those cocones are colimit cocones ... ??or something ??? ???something about case of x a kleisli category, or _something_ ??...

??consider "graphs with a unique edge" ... ?? =?= "bi-pointed sets" ??...

??hmm... "simplicial sets with a unique d-simplex" ... ????....

??well so, let's consider small categories ... ????as ... "freely adjoining colimits to the finitely presented (??os????) categories, except respecting the existing finite colimits" ... ???or something???

??ok, so i guess that i'm hopefully less confused now ... ???something about ... the locally finitely presentable category of models of a finite limits sketch can also be thought of as... ???the opposite of the free small limits theory on the
sketched finite limits theory ... ???or something ???.... ??rather than some sort of extra "duality twist" (or something...), as i was imagining for a bit ... ???

??so what about ... ?? "cocomplete category which is canonically the cocontinuous set-valued contravariant functors on it" ... ??or something ... ???something about "cocomplete category where representable pre-sheaf = pre-sheaf taking colimit cocones to limit cones" ... ???or something... so let's define a "continuous pre-sheaf" (on a small-ly cocomplete category???) to be one taking colimit cocones to limit cones ... ???then are the continuous pre-sheafs on a small-ly cocomplete category always precisely the representable ones??? ... ???maybe some "set-theoretic problems" even in just precisely formulating the question ??? ...