Consider the statement:
If is a fully faithful and essentially surjective functor, then it is an equivalence of categories.
If we use a set-theoretic foundation for mathematics, then this statement is equivalent to the axiom of choice. However, as we now show, in the internal logic of a regular 2-category, it is simply true. Thus, it should really be regarded as a “functor comprehension principle” or an “axiom of non-choice,” analogous for categories to the fact (true in any regular 1-category) that any injective and surjective function of sets is an isomorphism, or equivalently that one can always make “choices” when those choices are uniquely specified.
We start by characterizing full, faithful, and eso morphisms in the internal logic.
Consider first the following theory with two 1-types and :
Of course, the first axiom interprets by a morphism . This induces a map from over . The context of the second axiom above is the kernel pair of this map (as a map between discrete objects in ); thus the second axiom asserts that this kernel pair factors through the diagonal of , and hence is monic. But this is equivalent to saying that is faithful, so a model of this theory in a lex 2-category is precisely a faithful morphism.
Now consider the following theory, again with two 1-types and :
In a regular 2-category, the truth of this axiom asserts that the map is eso, which implies that is a full morphism. The converse is true at least in an exact 2-category. It is also true if is additionally faithful, since then is ff, hence (if also eso) an equivalence, so that is ff (and in particular full). Therefore, a model of the combined theory
in a regular 2-category is precisely a ff morphism.
In a merely lex 2-category, we can instead consider the theory
The first two axioms give a faithful morphism , as above, and now the third and fourth axioms supply a section of the map , which is thus both ff and split epic, hence an equivalence. Thus, a model of this theory in a lex 2-category is also precisely a ff morphism.
I don’t know whether there is a theory in lex 2-logic whose models in lex 2-categories are precisely full morphisms, but I suspect not. I also don’t know whether there is a theory in regular 2-logic whose models in non-exact regular 2-categories are precisely full morphisms.
Now consider the regular 2-theory
Since in context is interpreted by , the second axiom asserts that the image of the composite is all of , i.e. that this composite is eso. But this composite is clearly just , so a model of this theory is precisely an eso morphism.
Of course, we can then write the combined theory
whose models in exact 2-categories are precisely the eso+full morphisms.
It follows from the above considerations that a model of the following theory:
in a regular 2-category is precisely a morphism which is ff and eso, i.e. an equivalence. Since this theory evidently asserts that is “faithful, full, and essentially surjective on objects” in the internal logic, this is the desired “functor comprehension principle.”
Of course, in a Heyting 2-category, the latter three axioms can equivalently be stated as
which have perhaps the most familiar form.