A refactored proof of conceptual completeness

identity 1A, which is terminal in E/A, and the diagonal ∆A : A → A × A, which
is a new global element 1 → [A] ∈ E/A.
Given a model Ma : E/A → Sets we can immediately define
E M = [E] Ma a = Ma(∆A) : 1 → [A] Ma (i.e., a ∈ A M ).
On the other hand, an element a ∈ A M allows us to define a functor Ma by
sending each σ : E → A to the fiber (σ M ) −1 (a). A little thought shows that
these constructions are mutually inverse.
Whatever Th(M) is, it should at least have one model M∗ : Th(M) → Sets,
where we interpret each new constant ca as a itself. Moreover, that element a
should induce an interpretation ã : E/A → Th(M) such that M∗ ◦ã = Ma. This
suggests that Th(M) might be defined as a colimit of slice categories.
More generally, each σ : A → B induces a pullback functor σ ∗ : E/B → E/A.
Because pullbacks preserve fibers, whenever a ∈ A M and σ M (a) = b,
Mb ∼ = Ma ◦ σ ∗ : E/B → Sets.
Thus pullbacks will be the transition morphisms in our colimit.
The index category is provided by the Grothendieck construction, which
allows us to re-encode M as a “category of elements” M over E. An object
of M is a pair 〈A, a〉 where A is a formula (i.e., A ∈ E) and a is an element
of the definable set A M . We write 〈a ∈ A M 〉 for such an object. A morphism
〈a ∈ A M 〉 → 〈b ∈ B M 〉 is an arrow σ : A → B with σ M (a) = b. Composition is
computed in E, so we have an obvious projection M → E.
Using the fact that M preserves finite limits, one can show that M is a
filtered category:
• For any two objects 〈a ∈ A M 〉, 〈b ∈ B M 〉 there is a span
〈a ∈ A M 〉
p1
〈a, b〉 ∈ (A × B) M
4
p2
M 〈b ∈ B 〉