Cantor's theorem for presheaves? Some years back (before MathOverflow was born), Tom Leinster asked an interesting question at the $n$-Category Café which I don't recall ever seeing an answer for: 

Does there exist a category $C$ that admits an essentially surjective functor $F: C \to Set^{C^{op}}$? 

Terminology: we say that a functor $F: C \to D$ is essentially surjective 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, 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 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. 
 A: 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 $G(\alpha)$).

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.
A: I apologize in advance for my probable mistake's.
I did a proof for a small category. 
Let $\mathcal{C}$ any category, pose $\mathcal{C}^>:=Fun(\mathcal{C}, Set)$. If $\mathcal{C}$ has only one object $\ast$, let $G$ the monoid of its morphisms, then a presheaf $P\in \mathcal{C}^>$ is a $G^{op}$-Set  $G\times S_P\to S_P$.
Of course there are $G^{op}$-Set's of any cardinality, then cannot exist a essentially surjective small family of functor $F_i: \mathcal{C}\to \mathcal{C}^>\ i\in I$ (i.e. $\forall P\in \mathcal{C}^>\ \exists i\in I, X\in \mathcal{C}: F_i(X)\cong P$).
Suppose that $\mathcal{C}$ a (small) set of object, and  $F: \mathcal{C}\to \mathcal{C}^>$ is a essentially surjective functor, make $\widetilde{\mathcal{C}}$  identifying in $\mathcal{C}$  all object in only one $\ast$ and let $q: \mathcal{C}\to \widetilde{\mathcal{C}}$ the natural functor quotient. Let $P\in \widetilde{\mathcal{C}}^>$ see as a $G^{op}$-Set as above. Let $X_P\in \mathcal{C}$ with a isomorphism $\phi: F(X)\cong P\circ q$, then $F(X_P)$ is a family of set $F(X_P)(Y)\ Y\in \mathcal{C}$ with isomorphism's $\phi_Y: F(X_P)(Y)\cong S_{P}$  and this system is coherent with the morphism's  $F(X_P)(f): F(X_P)(Y)\to F(X_P)(Z)$ for $f: Z\to Y$ and the associate $q(f): \ast\to\ast$, and each of this morphism has a  "translate" $f_{X_P}:= {\phi_{X_P}}^{-1}\circ \phi_Z\circ F(f)\circ {\phi_Y}^{-1}\circ \phi_{X_P}$ as endomorphism of $F(X_P)(X_P)$, and this make $F(X)_P(X_P)$ a $G^{op}$-Set, with a $G^{op}$-coherent isomorphism $F(X_P)(X_P)\cong S_P$ i.e. a presheaf $\tilde{F}_P: \widetilde{\mathcal{C}}\to \widetilde{\mathcal{C}}^> $. 
If $\mathcal{C}$ is small  the couple's $(F(X)(X), \theta)$ where $X\in\mathcal{C}$ and $\theta$ is a $G^{op}$-action on $F(X)(X)$ form a (small) set, and this set contain any of the $G^{op}$-set $F(X_P)(X_P)$. Then we have a small family $\tilde{F}_i: \widetilde{\mathcal{C}}\to \widetilde{\mathcal{C}}^>\ X\in\mathcal{C}\ i\in I$ essentially surjective (absurd).
