For each topos $\mathbb E$ let $\mathcal O(\mathbb E)$ be the locally presentable category of objects in $\mathbb E$. We can make $\mathcal O$ into a contravariant functor to the category of locally presentable categories (with morphisms being cocontinuous) by assigning to each geometric morphism $(f^\\*, f\_\\*)$ the functor $f^\\*$. By [Mac Lane, Moerdijk: Sheaves in Geometry and Logic] this functor is representable, that is there is a topos $\mathbb A$, called the *object classifier*, such that there is a natural equivalence
$$
\mathrm{Hom}(\mathbb E, \mathbb A) \to \mathcal O(\mathbb E).
$$ 
Now I wonder whether $\mathcal O$ has a right adjoint, which I want to call $\operatorname{Spec}$ due to the analogy with algebraic geometry, that is whether there exists a contravariant functor $\operatorname{Spec}$ from the category of locally presentable categories to the category of topoi (with geometric morphisms) such that there is a natural equivalence
$$
\mathrm{Hom}(\mathbb E, \operatorname{Spec}\mathcal C) \to \mathrm{Hom}(\mathcal C, \mathcal O(\mathbb E))
$$
of categories.


(Here, *topos* shall mean *Grothendieck topos*.)



Topos associated to a category
This is described in the paper 



> 
> Bunge and Carboni, The symmetric topos, Journal of Pure and Applied Algebra 105:233-249, 1995.
> 
> 
> 


which describes the sense in which the construction you call Spec is analogous to the symmetric algebra construction. 


Bunge and Carboni give a biadjunction between the bicategory R, which is the opposite of the bicategory of Grothendieck toposes, and the bicategory A of locally presentable categories and cocontinuous functors (equivalently, left adjoints). 


