I am glad to see that a general question like [Is there a relationship between model theory and category theory?](https://mathoverflow.net/questions/11974/is-there-a-relationship-between-model-theory-and-category-theory "this") receives quite a lot attention and no down-votes for being too general and unspecific. So I feel encouraged to pose some follow-up questions (firstly restricted to first-order model theory).


### Preliminaries


I hope the following statements are sufficiently sensible, precise and correct.


* Each first-order theory $T$ with signature $\sigma$ unambigously defines a class of (ZF-)models.
* This class of models of $T$ together with the $\sigma$-homomorphisms form a category (the *category of models of $T$*).
* Two first-order theories with two arbitrary signatures may define [equivalent](http://en.wikipedia.org/wiki/Equivalence_of_categories) categories of models.



> 
> **Definition**: A first-order theory $T$
> *provides a model of a category $C$* if the
> category of models of $T$ is
> equivalent to $C$.
> 
> 
> 


* Each category $C$ defines a (possibly empty) set of first-order theories: the set of all $T$ which provide a model of $C$.


### Questions


*[Remark: I had to work this question over, since it seemed to be ill-posed.]*


Old version:
Can infinite [concrete categories](http://en.wikipedia.org/wiki/Concrete_category) be specified
other than as categories of (ZF-)models of some (possibly higher-order) theory? Examples?


New version (explicitly restricted to first-order theories):



> 
> Given an infinite category of models
> of a first-order theory $T$. Can this
> category - or one equivalent to it - be
> specified/represented/given
> independently of any first-order
> theory $T$ and its (ZF-)models?
> 
> 
> 


Remark: $T$ of course *can* be specified/represented/given independently of its models: as a set of formulas.




---



> 
> Why is the notion of *models of a (concrete)
> category* so uncommon? (Maybe because the
> answer to the first question is "No"?)
> 
> 
> 




---



> 
> Is there a genuine model-theoretic notion of two
> equivalent theories if these have
> two arbitrary signatures?
> 
> 
> 




---



Can infinite first-order categories be specified other than as categories of models?
Sure, by direct construction. Rings, preorders, the category of paths of a given graph, etc. But that's not what you wanted to know, is it?


