so.... ??direct sums of exact functors betweeen abelian categories are exact ... but kernels and cokernels of natural transformations between them generally aren't ... ??...

??so direct sums of flat functors from an algebroid to an abelian algebroid are flat ... ??...

??so maybe there's no particular good "tensor product" of abeian categories... ??...

i should probably ask toby bartels about some of this stuff...

so consider flat presheaves on the walking arrow quiver... vs flat reps of it... ??...

??so what _about_ topos / abelian category analogies here?? ... ??idea of "spectrum" of an abelian k-algebroid... exact functors into
the k-algebroid of k-vector spaces... (??and/or a stack of such functors into certain intended environments... ?? ...) ??something about karoubi-saturation here, but then beyond that as well ?? and so forth... ??...

?so what about the idea of "alexandroff abelian category" in analogy to concept of "alexandroff topos" (??in analogy to concept of "alexandroff locale" or something...)?? ... and so forth... ??...

??so what about the idea of "classifying topos for exact module of abelian ringoid x" ?? (or maybe introduce "base ring" here in certain way or ways ... something about ringed toposes... ringoid vs algebroid... module vs bi-module... and so forth...)

...and so forth... ??...