A refactored proof of conceptual completeness

this is equivalent to an element aH ∈ A MH .
(−)×A
MH
E Th(M)
˜∗
ã
H
E/A
MaH
Sets
From this we can define a family of functions hA : A M → A MH by setting
hA(a) = aH. If σ M (b) = a then ˜ b ◦ σ ∗ = ã. Consequently
H ◦ ã = H ◦ ˜ b ◦ σ ∗ = MbH ◦ σ∗ = MaH .
This ensures that hA ◦ σ M = σ MH ◦ hB, so that h is a natural transformation.
Thus the Th(M)-model H defines a model homomorphism h : M → MH. In
fact, this construction is reversible: h is sufficient to construct H.
Moreover, a natural transformation θ : H → K induces a homomorphism
MH → MK which commutes with the maps from M. This leaves us with the
following:
Proposition 2.1. Th(M) classifies E-models under M. If H : Th(M) → Sets,
H ◦ ˜∗ defines a model MH and the assignment h(a) = c H a defines an E-model
homomorphism h : M → MH.
3 Connecting syntax and semantics
In this section we will show that certain semantic properties of interpretations
I : E → F naturally correspond dual properties for the functors I∗ : Mod(E) →
Mod(F). Specifically, we will prove the theorem below (definitions to follow).
Theorem 3.1.
(a) I is conservative if and only if I ∗ is supercovering.
(b) I is full on subobjects if and only if I ∗ stabilizes subobjects.
(c) I is subcovering if and only if I ∗ is faithful.
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