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.



2-natural operations on toposes
The "2-natural operations" $U^n \to U$ correspond to functors $\mathbf{FinSet}^n \to \mathbf{Set}$. (edit: As Simon points out, these correspond to the *finitary* functors $\mathbf{Set}^n \to \mathbf{Set}$.)


The 2-functor $U$ is birepresented by the [object classifier](https://ncatlab.org/nlab/show/classifying+topos+for+the+theory+of+objects) $\mathbf{Set}[\mathbb{O}] = [\mathbf{FinSet},\mathbf{Set}]$ (the classifying topos for the theory of objects). Thus by the bicategorical Yoneda lemma, the category of pseudonatural transformations $U^n \to U$ is equivalent to the category $U(n\cdot \mathbf{Set}[\mathbb{O}])$ (where $ n \cdot {}$ denotes $n$-fold (bi)coproduct in $\mathrm{Topos}^\mathrm{coop}$), which is equivalent to the category $[\mathbf{FinSet}^n,\mathbf{Set}]$ (see Cole's paper [*The bicategory of topoi and spectra*](http://tac.mta.ca/tac/reprints/articles/25/tr25abs.html)). 


