The 2-categories of topoi and logoi

The category \(\Cat\) is cartesian closed (the functor \(C \times -\colon \Cat \to \Cat\) admits a right adjoint \(\Fun(C,-)\colon \Cat \to \Cat\) for all \(C\)) and thus enriched over itself. This gives rise to a 2-category \(\bbCat\), whose objects are categories, whose morphisms are functors, and whose 2-morphisms are natural transformations between functors: \(\Hom_{\bbCat}(C,D) = \Fun(C,D)\).

This in particular allows us to upgrade the category \(\Logos\) to a 2-category, by regarding it as a locally full subcategory of \(\bbCat\):

The Hom-categories \(\bbLog\) are given by the full subcategories

spanned by the morphisms of logoi. We may similarly upgrade \(\Topos\) to a \(2\)-category, by setting

In other words, we define the Hom-categories of \(\bbTop\) to be \(\Fun_{\bbTop}(T,S) := \Fun_{\bbLog}(S,T)\).