This is a brief note to record an interesting fact that I noticed recently.

Consider the following double category, which I will call \mathbf{Lex}.

- The objects are logoi (e.g., the categories of sheaves for topoi)
- The proarrows are left exact functors (functors which preserve finite limits)
- The arrows are geometric morphisms (e.g. left exact functors with right adjoints)
- The 2-cells are natural transformations between the squares of left exact functors

It is not hard to see that \mathbf{Lex} has companions; simply forget that a geometric morphism has a right adjoint! What is slightly more surprising (though I believe still known) is that \mathbf{Lex} also has conjoints! The conjoint to a geometric morphism is given by its right adjoint, which preserves all limits and so is in particular lex; the 2-cells in the definition of conjoint are given by the counit and unit of the adjunction!

So \mathbf{Lex} is an equipment. (What I donâ€™t know is whether it is a cartesian equipment, because Iâ€™m not so familiar with how coproduct topoi interact with lex morphisms.)

But according to @david.jaz this is all known (I donâ€™t know where it is known). The interesting fact that I think might be original is that Artin gluings are tabulators in \mathbf{Lex}!

Specifically, suppose that we have two logoi E and E', and a lex functor \Phi \colon E \to E'. The Artin gluing is a logos \mathbf{Gl}(\Phi) where an object is given by a tuple (e \in E, e' \in E', f \colon e' \to \Phi(e)). There is then a natural 2-cell

which satisfies the universal property of tabulator! I should give a detailed proof of this, but I should also do other things, so Iâ€™m just going to end this here.