Regarding the consistency condition of a natural transformation of database instances (set-valued functors), the field of relational database theory studies relational databases and their morphisms. In the relational context (or Rel-valued functor context), a morphism of databases/sets of relations must preserve 'facts'; i.e., h : I => J when I(x,y,...,z) implies J(h(x),h(y),...,h(z)). It turns out that this relational definition of 'consistency as fact-preservation', when restricted to databases comprised of total functions only, is exactly the naturality condition for transformations between set valued functors. So to summarize: if you were talking to a database theorist you could say 'fact preserving' instead of 'consistent' to describe the same concept.