Suppose that $\mathcal{T}$ is an abstract $2$-category we know is equivalent to the $2$-category of Grothendieck topoi via some equivalence $$\phi:\mathcal{T} \to \mathfrak{Top},$$ and let $E$ be an object of $T$. Can we recover the underlying category $\phi(E)$ without using $\phi$?
I am asking because often properties of morphisms of topoi use that we know what topoi are (certain categories) and what maps between them are (certain pairs of adjoint functors), e.g. by referencing elements of the domain, or by saying one of the pairs of adjoint functors has a further adjoint with certain properties. But, a general principle of category theory is that one shouldn't care what things are, just about the maps between them-and not really the maps, but how they are related, i.e. the category you get. If you have an equivalent category, then you should be able to make the same statements. So the $2$-category $\mathcal{T}$ should be enough. Hence, I ask, how does one recover $\phi(E)$? It would suffice have a completely categorical description of etale geometric morphisms (i.e. one not depending on the "evil" choice of a particular presentation of $\mathfrak{Top}$ as consisting of certain categories called topoi) because any topos is equivalent to the category of etale morphisms over itself.
Best Answer
There is a Grothendieck topos $\textbf{Set}[\mathbb{O}]$ with the following universal property: for all Grothendieck toposes $\mathcal{E}$, the category $\textbf{Geom}(\mathcal{E}, \textbf{Set}[\mathbb{O}])$ of geometric morphisms $\mathcal{E} \to \textbf{Set}[\mathbb{O}]$ and "geometric transformations" (a misnomer; they actually code algebraic data!) is naturally equivalent to $\mathcal{E}$ itself. Such a $\textbf{Set}[\mathbb{O}]$ is called an object classifier.
We can construct $\textbf{Set}[\mathbb{O}]$ explicitly using the theory of classifying toposes: one presentation is as the presheaf topos $[\textbf{FinSet}, \textbf{Set}]$. Indeed, by Diaconescu's theorem, a geometric morphism $\mathcal{E} \to [\textbf{FinSet}, \textbf{Set}]$ is the same thing as a left exact functor $\textbf{FinSet}^\textrm{op} \to \mathcal{E}$, but any such is freely and uniquely determined by the image of $1$; this correspondence extends to 2-morphisms as well.
Addendum. To address Simon Henry's comments, here is an abstract construction of $\textbf{Set}[\mathbb{O}]$. It is known that the 2-category of Grothendieck toposes has tensors with small categories. Indeed, $$[\mathbb{C}, \textbf{Geom}(\mathcal{E}, \mathcal{F})] \simeq \textbf{Geom}([\mathbb{C}, \mathcal{E}], \mathcal{F})$$ but we know that $\textbf{Set}$ is a pseudo-terminal object in the 2-category of Grothendieck toposes, so we may take $\textbf{Set}[\mathbb{O}] = \textbf{FinSet} \otimes \mathcal{S}$, where $\mathcal{S}$ is any pseudo-terminal Grothendieck topos.