Reconciling statements about accessible categories

Question

I have a problem reconciling the following assertions regularly made online about accesssible categories:

1. The category $\mathrm{Mod}(T)$ of models of some first-order theory $T$ with countable signature is $\aleph _{1}$-accessible. (e.g. stated in the Wikipedia article on accessible categories)

2. A small category is accessible precisely when all idempotents split. (e.g. stated in the nLab article on accessible categories)

From the second assertion it follows that the category $\mathcal{C}$ with one object and one non-trivial idempotent endomorphism, is not acccessible.

Now, take the first-order theory $T$ with one constant symbol $c$, and two axioms:

• $\forall x. \forall y. \forall z. x = y \vee x = z \vee y = z$,
• $\exists x. \exists y. x \neq y$.

The signature of $T$ is countable, so by the first asssertion, it follows that $\mathrm{Mod}(T)$ is accessible. But what I'd normally call the category $\mathrm{Mod}(T)$ (objects: models of $T$ and morphisms: $c$-homomorphisms) is not accessible: the theory is categorical and the only model has one non-trivial endomorphism (sending everything to $c$), so the skeleton of $\mathrm{Mod}(T)$ is in fact isomorphic to $\mathcal{C}$ defined above.

Question: What am I missing?

I know how to prove Assertion 2 from the definition of acccessible categories, so my guess is that something's wrong with (my interpretation of) Assertion 1. Maybe $\mathrm{Mod}(T)$ is intended to refer to a different category (One with different morphisms? Elementary embeddings?), or to a restricted notion of theory $T$.

Answer 1

In Chapter 5 of Locally presentable and accessible categories it is shown that the category of models of a first order theory and elementary embeddings is accessible. The proof uses the Löwenheim–Skolem theorem. It is not explicitly calculated, but it should be the case that the category is $\aleph_1$-accessible if the signature is finitary and the language is countable.

I am not sure why the focus is on elementary embeddings. Perhaps it has to do with the practice of model theory. However, when working with models of a theory in a more restricted fragment of logic, say geometric logic, category theorists do in fact usually consider the category of models and homomorphisms.