A refactored proof of conceptual completeness

Since T is consistent we may find an F-model N0 and an element a0 ∈ S N0
such that a0 ∈ IR N0 for each R ∈ Γ. This means that I ∗ N0 and a0 satisfy the
assumptions in the second part of the lemma, so that TI ∗ N0
is consistent.
A model of the latter theory consists of another F-model N1 together with
an E-model homomorphism h : I ∗ N0 → I ∗ N1 such that h(a0) ∈ S N2 . Then
a0 ∈ S N1 witnesses the fact that I ∗ does not stabilize S.
3.10 Proof of (c)
Definition 3.11. We say that I : E → F subcovers B ∈ F if there is an object
A ∈ E, a subobject S ≤ IA ∈ F and a regular epimorphism q : S ↠ B. I is
subcovering if it subcovers every B ∈ F.
Fix an object B ∈ F. Given an object A ∈ E, a partial function IA ⇀ B is
a two-place relation ϕ ↣ IA × B such that
F/(〈b, b ′ 〉 : B × B) ⊲ ϕ(a, b) ∧ ϕ(a, b ′ ) ⊢a:IA b = b ′ .
Categorically speaking, ϕ is a partial function just in case the projection ϕ ↣
IA × B → IA is monic.
We sometimes write ϕ : IA ≥ S → B to indicate that S = (∃b : B.ϕ) is the
domain of definition for ϕ. Now define
Φ = {ϕ ↣ IA × B | A ∈ E, ϕ : IA ⇀ B}.
Lemma 3.12. If I does not subcover B and F/B is consistent then:
(i) the theory T = F/(b:B) ∪ {∃a:IA. ϕ(a, b) ⊢ ⊥ | ϕ ∈ Φ} is consistent.
(ii) for any T-model 〈N, b0〉 the following theory TN is consistent:
Th 1 (N) ⊕
F⊕Th(I E
∗N) Th 2
(N) ∪ {c 1 b0 = c2b0 ⊢ ⊥}
12