so is there a barr-beck distributivity between the "finite limits" and "finite colimits" monads on the 2-category of algebroids (or something like that) that gives rise to the "abelian category" monad?

taking "op-modules" is freely adjoining colimits... taking the opposite of the modules algebroid is freely adjoining limits...

so let x be an algebroid, and consider the algebroid of fp modules of the fp module algebroid of x, compared to the algebroid of fp modules of the fp op-module algebroid of x. are these algebroids opposite to each other??

fp modules of the fp right-module algebroid, compared to fp modules of the left-module algebroid...

so let f be an fp module of the fp module algebroid of x. then let's try to define an fp module g of the fp op-module algebroid of x as follows:

g evaluated at an fp module m of x^op should be ... m tensored over something with something ... or something ...

hmm, perhaps the point is that there should be some very straightforward morita equivalence here; let's try to make that explicit...

or wait a minute... is the alleged morita equivalence here contravariant, and does that actually make any sense??....

hmm, perhaps this (the existence of the invertible distributivity natural functor here) is obvious if we believe the alleged description of free abelian categories that we think we heard...