what about modules of the algebroid of coherent sheaves over a nice variety or stack (or something), vs some sort of sheaves of modules of module algebroids... ?? or something...

??what about simultaneously extending tensor product by usual distributivity over colimits, and limits by "abelian cat distributivity"?? or something?? ... ??something about "extending distributivity by distributivity" or something???....

??what about algebraic-geometric morphisms with algebraic part preserving limits?????? and so forth, or something... ??also something about "flatness" ....

what about "toric case" of some stuff here??...

what about stuff here (or something...) and "symmetric tensors ct polynomials" (or something...) and "strongly n-dimensional object" ... ?? ... or something... ???? ... ????something about "the doctrine of all multi-additive functorial operations on the algebroid of vector spaces of a finite field" ... ??and so forth...



??what about freely adjoining limits and/or colimits to a symmetric monoidal category, and also algebroidizing it??? ... or various sub-sets of that or something... ??? ...

??what about abelian group objects in a symmetric monoidal complete category?? whether there's a good notion of tensor product of them??? ...

??what about stuff here (or something...) and "schur functors" (or something...)? ...