?? so ... ?? we really need to try this idea again ... of thinking about (categorified ...) "presentation" of (?? non-toric and / or toric ... ??? ...) quasicoherent sheaf category as cocomplete (?? enriched or otherwise ... ?? ...) cocomplete category ... and using it to understand "correspondence" in sense of left adjoint; then trying to relate this to "correspondence" in sense of bi-quasicoherent sheaf (?? and / or related ideas ... ?? ...) .... ??? possible relationship to "absolute colimit with a twist ..." ?? ... ??? ...

?? vague stab ... ?? in 1 -|-> a -|-> b situation ... c on left, d on right ... ?? "inflate c to aXb, then tensor with d, and also with diagonal ... then finally take global sections ..." ??? .... ???? ...

?? vague (?? ...) idea of [absolute colimit] as related to ["getting homming to work nicely"] ... ???? .... ?? discrete case (?? where maybe it's less vague ?? ... "commutative algebraic theory" ... operations as also homomorphisms ... ?? interchange laws ?? ... ??? ...) vs beyond ?? .... ???? .... "matrrix formalism" .... ??? .... "correspondence" ... "glueing" ... input vs output ... ???? ....

?? matrix-monoidal ag correspondence ... ??? ... category ( / gpd ?? ...) as "matrix-monoid" ... ??? .... ??? .... ?? "stack" ?? .... ??? ..... ?? relationship to other flavors of co- / monoid structure on correspondence ?? ... ?? ... ??? feyman diagrams ?? .... ???? .....

?? stopped to wonder whether some sort of "you can't play both sides of gabriel-ulmer duality" result might cause annoyance with some of this ag correspondence stuff ... ??? but at the moment i'm having some trouble even trying to get such a result to sound like a real hindrance (?? ... or "surprise" ... ?? ... ??? ...) in any circumstance .... ?? ...

??? thinking about poset case .... ?? .... large posets ??? ... ??? cocomplete such, and their duals ???? .... ???? ..... .... ????? .....