To be clear, I mean the category $\mathbf{Topoi}$ whose objects are all elementary topoi, and whose morphisms are logical functors between topoi. Does this category itself form a topos? What about the category $\mathbf{Groth}$ of Grothendieck topoi (with geometric morphisms)? If yes, do either of these categories form a Grothendieck topos?
I see that $\mathbf{Cat}$ does not form a topos.