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*.)

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