Model-theoretic interpretations seem to give rise to categories in which morphisms are not functions. To name just two small examples:

- the category of finite graphs with interpretations between them
- the category of finite relational structures of arbitrary signature with interpretations between them

In the category of finite graphs with graph homomorphisms one is primarily interested in the finitely many *in*-going morphisms of an object (which tell all there is to know about a given object). In an interpretational category there are infinitely many in-going morphisms (= interpretations), but only finitely many *out*-going ones: to those objects (= structures) a given object is able to interpret. So maybe the opposite category might be better suited for comparison's sake?

Maybe I am blind, but I didn't manage to find any thorough analysis of such a category with interpretations as morphisms. Maybe I just missed the right keywords to search with?

Where can I learn more about categories with interpretations as morphisms?

**Addendum:** I should have mentioned the reminiscence of homotopy when thinking of interpretations this way:

Homotopy categories are standard examples of categories with morphisms that are not functions.

In the Wikipedia article on interpretations we read: "[Bi-interpretability] permits to define an equivalence relation among structures, reminiscent of the homotopy equivalence among topological spaces."

In chapter

*5.4 Shapes and sizes of interpretations*of his*Model Theory*, Wilfrid Hodges uses the term "homotopy" in the context of*(c) Homotopies and bi-interpretations*(but the connection with the standard definition of a homotopy isn't clear to me, to be honest).