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 $\mathbf{Set}[\mathbb{O}] = [\mathbf{FinSet},\mathbf{Set}]$ (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).

(For me the category of toposes is the opposite of the category of left exact left adjoint functors and natural transformations, so $Topos^{co}$ in your sense)

The functor $U$ is representable by the classifying topos of objects, i.e. the topos $S[\mathbb{O}]$ which as a category is the category of functors from finite sets to sets, i.e. :

$$ U( \mathcal{T}) = Hom(\mathcal{T}, S[ \mathbb{O}] ) $$ Similarly,

$$ U(\mathcal{T}) ^n = Hom(\mathcal{T}, S[\mathbb{O}]^n)$$

$S[\mathbb{O}]^n$ being the category of functors from (finite set)$^n$ to Set.

Now by the Yoneda lemma, and up to $2$-categorical details that I will totally ignore, a natural transformation from $U^n$ to $U$ is the same as a morphism from $S[\mathbb{O}]$ to $S[\mathbb{O}]^n$, i.e. it is given by an object of $S[\mathbb{O}]^n$, i.e. a functor from (finite set)$^n$ to Set.

Claim/exercice: given a functor from (finite set)$^n$ to $Set$ its actions $Set^n \rightarrow Set$ corresponds to the unique extension commuting to directed colimits. So in the end those "operations" are exactly the same as the operations $Sets^n \rightarrow Sets$ which commutes to directed colimits.

I believe everything works exactly the same for $\infty$-toposes, replacing sets and finite sets by "spaces" and "finitely generated spaces" (I mean finite CW-complexes), (Of course this will relies on a large amount of results from Lurie's books, although I think one can avoid manipulating $(\infty,2)$-categories by just forgeting the non-invertible $2$-cells at least in a first time...)