Some years back (before MathOverflow was born), Tom Leinster [asked](https://golem.ph.utexas.edu/category/2008/01/2toposes.html#c014438) an interesting question at the $n$-Category Café which I don't recall ever seeing an answer for: 



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> Does there exist a category $C$ that admits an essentially surjective functor $F: C \to Set^{C^{op}}$? 
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Terminology: we say that a functor $F: C \to D$ is [essentially surjective](http://ncatlab.org:8080/nlab/show/essentially+surjective+functor) if every object $d$ of $D$ is isomorphic to some value $F(c)$. This is the good notion of surjectivity for the 2-category $Cat$, or at least one good notion. 


As is well-known from categorical circles, the presheaf category $Set^{C^{op}}$ here plays a role of "power object" $P(C)$ that is usefully regarded as analogous to power sets in set theory or more generally in toposes. (For example, the Yoneda embedding $y\_C: C \to Set^{C^{op}}$ plays a role analogous to the singleton mapping $\{-\}: S \to P(S)$ from set theory.) In fact Tom's question is embedded in a larger discussion of what one should mean by a '2-topos' -- see that discussion for more on the analogy. 


So the question above is analogous to one that Cantor's theorem answers: can one have a set $S$ that maps onto its power set $S$? So the expected answer to the question is '**no**'. Note however that the standard diagonalization technique behind Cantor's theorem, as explained for example [here](http://arxiv.org/abs/math/0305282), doesn't apply in any obvious way since there is no general decent notion of diagonal map $C \to C \times C^{op}$. 


Regarding foundational issues: I'll leave that up to you. :-) If you want me to impose a constraint, we might add the condition that $C$ is locally small, but note that we'll soon be leaving the land of local smallness anyway, since there is a [result](http://www.tac.mta.ca/tac/volumes/1995/n9/1-09abs.html) due to Freyd and Street that if also $Set^{C^{op}}$ is locally small, then $C$ is (equivalent to) a small category, and that would be a *huge* constraint that makes the question not so interesting. 



Cantor's theorem for presheaves?
No such category exists. My original argument for this assumed local smallness and is below the break; here is a simpler argument that does not require local smallness (though it does basically use my original argument in the special case $\mathbf{C}=\mathbf{Set}$).


Let us take $\kappa$ to be an inaccessible cardinal and work with $V\_\kappa$ as our universe, so the categories $\mathbf{Set}$ and $\mathbf{C}$ we start with are classes in $V\_\kappa$ (so $\mathbf{Set}$ is the category of sets in $V\_\kappa$), and we only go outside $V\_\kappa$ to form functor categories. Suppose that there is an essentially surjective functor $\mathbf{C}\to\mathbf{Set}^{\mathbf{C}^{op}}$ and let $H$ be the associated functor $\mathbf{C}\times\mathbf{C}^{op}\to\mathbf{Set}$. I will obtain a contradiction by proving that there are $2^\kappa$ non-isomorphic objects of $\mathbf{Set}^{\mathbf{C}^{op}}$.


First of all, for every cardinal $\lambda<\kappa$, there is a constant presheaf $\lambda$ on $\mathbf{C}$, and so there is some object $A\_\lambda$ with the property that $|H(A\_\lambda,B)|=\lambda$ for all $B$. Now fix any object $B$, considered as an object of $\mathbf{C}^{op}$. Write $G(A)=H(A,B)$; then $G$ is a functor $\mathbf{C}\to\mathbf{Set}$ with the property that $|G(A\_\lambda)|=\lambda$ for all $\lambda$. Consider $G$ as a functor $G^{op}:\mathbf{C}^{op}\to\mathbf{Set}^{op}$. We can then compose $G^{op}$ with any functor $P:\mathbf{Set}^{op}\to\mathbf{Set}$ to get a new presheaf $PG^{op}$ on $\mathbf{C}$. I claim that there are $2^\kappa$ choices of $P$ which give rise to non-isomorphic presheaves $PG^{op}$.


Indeed, since $G^{op}$ is essentially surjective, it suffices to give $2^\kappa$ different functors $P$ such that the induced maps $\{\text{cardinals }\lambda<\kappa\}\to\{\text{cardinals }\lambda<\kappa\}$ are distinct. This is not difficult; for instance, it can be done by a variant of the "wedge of spheres" construction below (let $\mathbf{C}=\mathbf{Set}$, $F(A)=|A|$, and instead of just taking a single copy of each sphere when constructing $T(Q)$, add enough spheres to change the cardinality of $T(Q)$ at $G(\alpha)$).




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Let's work in the context of Grothendieck universes and require our categories to be locally small. Then I claim that no such category exists.


Let $\kappa$ be an inaccessible cardinal and let $V\_\kappa$ be our base universe. Let $\mathbf{C}$ be a locally small category. Define an *exhaustion* of $\mathbf{C}$ to be an unbounded function $F:\operatorname{Ob}(\mathbf{C})\to \kappa$ such that if $B$ is a retract of $A$ then $F(B)\leq F(A)$.


First, I claim that if $\mathbf{C}$ has an exhaustion $F$, then $\mathbf{Set}^{\mathbf{C}^{op}}$ has $2^\kappa$ non-isomorphic objects and hence there is no essentially surjective functor $\mathbf{C}\to \mathbf{Set}^{\mathbf{C}^{op}}$. Let $1$ be the constant singleton presheaf on $\mathbf{C}$; let $\*\_B$ denote the unique element of $1(B)$ for all objects $B$. Given an object $A$ of $\mathbf{C}$, let $S^A$ (the "$A$-sphere", by analogy with the case $\mathbf{C}=\Delta$) be the presheaf obtained from $1$ by freely adjoining an element of $S^A(A)$ whose image under every map $A\leftarrow B$ is $\*\_B$ for all $B$ such that $F(B)<F(A)$. Since $A$ is not a retract of any such $B$, this new element of $S^A(A)$ will not be equal to $\*\_A$.


Now let $I\subseteq \kappa$ be the image of $F$ and choose a right inverse $G:I\to\operatorname{Ob}(\mathbf{C})$ of $F$. For each $Q\subset I$, define $T(Q)$ to be the colimit of the diagram consisting of the inclusions $1\to S^{G(\alpha)}$ for all $\alpha\in Q$. This colimit exists because for any object $A$, $1\to S^{G(\alpha)}$ is an isomorphism at $A$ for all $\alpha>F(A)$, and hence this colimit is small at $A$. We can determine the set $Q$ from the presheaf $T(Q)$ the same way you can determine the non-degenerate simplices of a simplicial set. Thus the presheaves $T(Q)$ are all non-isomorphic. Since there are $2^\kappa$ different values of $Q$, this proves the claim.


To prove the claimed theorem, it now suffices to show that any essentially large locally small category has an exhaustion. Let $\mathbf{C}$ be an essentially large locally small category, and assume WLOG it is skeletal. By essential largeness, let $f:\operatorname{Ob}(\mathbf{C})\to \kappa$ be a bijection. By local smallness, each object of $\mathbf{C}$ has fewer than $\kappa$ other objects as retracts (since a retraction is determined by the associated idempotent endomorphism). We can thus define $F:\operatorname{Ob}(\mathbf{C})\to \kappa$ by $$F(A)=\sup \{f(B):B\text{ is a retract of }A\},$$
and this $F$ will be an exhaustion.


