Does the 2-category of Grothendieck topoi have exponential objects?


There are size issues: Since Grothendieck topoi are supposed to have a small set of generators, the collection of objects of a Grothendieck topos has to be a *class* and cannot be a [conglomerate](https://en.wikipedia.org/wiki/Conglomerate_(mathematics)). However, classes (in contrast to conglomerates) don't have nice closure properties: if $A$ and $B$ are classes, then the conglomerate $B^A$ is, in general, *not* a class.


So for the purposes of my question, drop the small set of generators from the Giraud axioms. :-)



Does the 2-category of topoi have exponential objects?
No. [Some, but not all topoi are exponentiable](https://ncatlab.org/nlab/show/exponentiable+topos).


