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.

Addendum. To address Simon Henry's comments, here is a 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.

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.

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.

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.

Addendum. To address Simon Henry's comments, here is a 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.