One of my friends asked me whether or not the inclusion of the category of Grothendieck toposes into elementary toposes has a left adjoint. We are looking at the categories of geometric morphisms. I am not really sure how to start but nothing seems to rule it out immediately. 



Is there a "Grothendieckification" functor from elementary toposes to Grothendieck toposes?
No, it doesn't. If it did, then it would preserve limits. But the category of Grothendieck toposes and geometric morphisms has a terminal object, namely the category of sets, while there are elementary toposes not admitting any geometric morphism to Set (for instance, any small elementary topos).


