A refactored proof of conceptual completeness

These logical doctrines often correspond to certain categorical constructions.
Finite products are sufficient to express pairing and equality, allowing us to
interpret algebraic theories. Quantifiers can be interpreted as adjoints to sub-
stitution, and in a topos we can use subobject classifiers and exponentials to
interpret higher-order constructions. Functors which preserve these structures
are interpretations or models of the theory.
We will be particularly interested in the doctrine of pretoposes. This is
almost, but not quite, a fragment of first-order logic. Pretoposes have all finite
limits and regular factorizations, providing interpretations for ⊤, ∧, ∃. To these
we add two new connectives/type constructors for disjoint sums and quotients
by equivalence relations.
Quotients and finite coproducts are definable in Sets. Consequently, if
T → T extends T by adding quotients and coproducts, then any T-model M
has a unique extension to a T-model M. This, together with completeness
of set models, suffices to ensure that the addition of sums and quotients is a
conservative extension of T.
The principle benefit of adding these additional types is that it allows for
a more robust notion of notion of interpretation E → F. Thinking of F as a
semantic context, we now allow ourselves to construct E-models by quotienting
definable equivalence relations.
The unique extension of T-models to T-models also shows that the category
of models cannot possibly distinguish between T and T. The conceptual com-
pleteness theorem of Makkai and Reyes says that this worry goes no further: a
pretopos is determined (up to equivalence) by its category of models.
Specifically, suppose that we have a pretopos functor I : E → F. Given a
model N : F → Sets, we can “restrict along I” to define an E-model I ∗ N =
N ◦ I : E → Sets.
Theorem (Conceptual Completeness). A pretopos functor I : E → F is an
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