so given a ringoid r (perhaps restricted to be finitely presented or something like that?) and a fp _fp r-module_-module, can we give a corresponding syntactic construction in the doctrine of abelian categories? and so forth...

then consider, instead, a ringoid equipped with, for each n-tuple x of ringoids and each fp _fp x-n-module_-module, a corresponding multi-functorial operation...

back to the case n=1...

r ringoid (??perhaps fp or something??)

m fp _fp r-module_-module

x ab cat

f : r -> x

then we want to obtain an object of x ...

??for each generator g of m, evaluate f at the object where g lives ... ??

??for each relator r of m, evaluate f at the morphism where r lives ... ??

??take the colimit in x of the diagram ... ??...

???is this some sort of weighted colimit of the diagram f with weighting given by ... ?? ??or something like that??

wait a minute, i think that there are some level slips or something here...

let's go back and try specializing to some example that we think we can understand...

r is the walking object; that is, the ring z as a one-object ringoid.

m has one generator g at the z-module z, and one relator r = "g*2".

x is some abelian category, and f is essentially some object of x.

the resulting object of x is, we think, the cokernel of the scalar 2 acting on the object f ... or something like that ...

perhaps we want to kan-extend f... and then kan-extend the kan-extension the other way... ??...