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.)
Best Answer
This is described in the paper
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).