Any pseudonatural endomorphism $\Phi$ of the forgetful 2-functor $U:Topos^{coop}\to Cat$ is essentially determined by its component $\Phi_{Set}$. But which endofunctors of $Set$ induce such a $\Phi$? More generally, one can consider pseudonatural transformations $U^n \Rightarrow U$, which are determined by a functor $\mathsf{Set}^n \to \mathsf{Set}$.

Here

$\mathsf{Topos}^\mathrm{coop}$ is the 2-category of Grothendieck toposes and left exact left adjoint functors -- doubly dual to the 2-category of toposes and geometric morphisms.

The functor $U : \mathsf{Topos}^\mathrm{coop} \to \mathsf{Cat}$ sends a topos to its underyling category and a left exact left adjoint to its underlying functor.

For $n \in \mathbb N$, the functor $U^n$ is the composite of $U$ with the $n$th power functor $\mathcal C \mapsto \mathcal C^n$.

So it's natural to define a *2-natural operation on toposes* to be a pseudonatural transformation $U^n \to U$ for $n \in \mathbb N$. In this language, the question is:

**Question:** What are all the 2-natural operations on toposes?

To see that such an operation $\Phi: U^n \Rightarrow U$ is determined by the component $\Phi_\mathsf{Set}: \mathsf{Set}^n \to \mathsf{Set}$, first note that $\Phi$ is determined by its components on presheaf toposes, because left exact localizations are essentially split epimorphisms in $\mathsf{Cat}$. Then the components at presheaf toposes are determined by the components at $\mathsf{Set}$ because $U$ preserves $\mathsf{Cat}$-cotensors.

I would also be interested in the analogous question for $\infty$-toposes.