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 uniquely determined by the image of $1$; this correspondence extends to 2-morphisms as well.