This question is related to this one: Local smallness and (higher) topoi which has not yet been answered.
The $2$-category of topoi and geometric morphisms is not locally small. (As I mentioned in the question above, for example, if $A$ is the classifying topos for abelian groups, the category of geometric morphisms from $Set$ to $A$ is equivalent to the category of all abelian groups, which is not small.)
**Question:**Is the $2$-category of topoi and only etale geometric morphisms locally small?
Note: I'm actually interested in the corresponding statement for infinity topoi.
Remark: Notice that this $2$-category is locally a topos, so, it is locally locally small (sorry for the weird wording). That is, if $E$ is a topos, the slice 2-cat over $E$ consisting of etale morphisms over $E$ is known to be equivalent to $E$ itself (and hence is really a 1-category). There is a corresponding result for infinity topoi as well. Perhaps there is a way to use this to prove the result.