Hi Matthew, I'm happy to move the discussion to another thread - if you start one, let me know.
Just to answer your question, the STLC forms a cartesian closed category, where the objects are types and the morphisms are terms. Similarly, the category of categories forms a cartesian closed category, where the objects are categories and the morphisms are functors. Similarly, AQL schemas form a cartesian closed category, where the objects are schemas (finitely presented categories) and the morphisms are schema mappings (morphisms of finitely presented categories). That is why it is natural to translate STLC types <-> categories and STLC terms <-> functors.
Under the above analogy, a concrete example of a finitely presented category that I'm not sure how to represent as a Haskell type/type family/etc but I know how to represent as a Coq type, is the presentation with two objects, A and B, two generating arrows f : A -> B and g : B -> A, and the equation f.g = id.