so by analogy with my understanding of the role of the concept of "flat presheaf" in diaconescu's theorem, i'm guessing that a "flat module" of a k-algebroid r should be... what?? ...

we're trying to understand the left-universal property of the k-algebroid of fp r-modules, as an abelian k-algebroid... we start out by understanding its left-universal property as a finitely co-complete k-algebroid... which is that it's the "walking r^op-module" (or perhaps "walking r^op-shaped diagram"...) ... so an exact k-functor from the k-algebroid of fp r-modules to an abelian k-algebroid x is given by kan-extending an r^op-shaped diagram in x which has the special property that the kan extension (?which is the process of "tensoring" an fp r-module with the r^op-shaped diagram...) preserves kernels.

??a flat k-functor r->x is the restriction along the yoneda embedding r->_r^op-module_ of an exact k-functor _r^op-module_->x ... ??...