TITLE: When is the category of small (pre)sheaves a(n elementary) topos?
QUESTION [10 upvotes]: When $C$ is essentially small, the presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is the free cocompletion of $C$. The presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is also a topos.
When $C$ is truly large, the presheaf category $[C^\mathrm{op},\mathsf{Set}$] is not even locally small. But the full subcategory $\mathcal{P}C \subset [C^\mathrm{op},\mathsf{Set}]$, consisting of the presheaves which are small colimits of representables, is still the free cocompletion of $C$. When $C$ is small, $\mathcal{P}C$ is the whole category $[C^\mathrm{op},\mathsf{Set}]$, so it is a topos. But when $\mathcal{C}$ is large, is $\mathcal{P}C$ a topos? 
Day and Lack, following Rosicky, study when $\mathcal{P}C$ is complete, and when it is cartesian closed. Certainly cartesian closedness is necessary to be a topos; I believe that a cocomplete topos is also complete, so that completeness is also necessary. But I suspect that $\mathcal{P}C$ might only have a subobject classifier if $\mathcal{C}$ is essentially small.
If it makes a difference to restrict to sheaves in some Grothendieck topology, that would be interesting to know. I suppose my motivating example is when $C$ is the category of schemes (or just affine schemes). Algebraic geometers are very interested in sheaves on this category in various topologies; I suspect most such interesting sheaves are small, but I don't know if that restriction gives one a topos to work with.
EDIT
As David Carchedi points out in the comments, I should emphasize that I'm interested in when $\mathcal{P}C$ is an elementary topos (a finitely complete, cartesian closed category with a subobject classifier). As Eric Wofsey argues in the comments, if $C$ is essentially large, then $\mathcal{P}C$ is never a Grothendieck topos (it doesn't have a small generating set).
EDIT
The weakest of observations: the Yoneda embedding $C \to \mathcal{P}C$ is still continuous, so preserves monos, and is fully faithful; so it induces injections on subobject lattices. So if $C$ is not well-powered to begin with, then $\mathcal{P}C$ is also not well-powered, so can't be a topos (given that it's locally small: look at the size of $\mathrm{Hom}(a,\Omega)$).
EDIT
It looks like the canonical way to give a general answer to this problem would be to generalize the work of Menni which characterizes when the exact completion of a finitely complete category is a topos. Taking the category of small sheaves is put in the same framework as exact completions in Mike Shulman's paper which Adeel linked to in the comments. So the way forward ought to be clear. I'm not sure how easy criteria along Menni's lines would be to check in particular cases like schemes, though.

REPLY [5 votes]: As you seem to suspect, $\mathcal{P}(C)$ is rarely a topos if $C$ is large. As a simple example, if $C$ is large discrete (identified with a large set $X$) and $\mathbf{Set}$ is the category of small sets, then $\mathcal{P}(C)$ is equivalent to $\mathbf{Set}/X$, which does not have a terminal object. 
You are correct that a cocomplete topos $E$ is complete. For that we would just need small products. If $\{X_i\}_{i \in I}$ is a small family of objects of $E$, then under cocompleteness there is an obvious fibered object $p: \sum_i X_i \to \Delta I$ where the codomain $\Delta I$ is the coproduct of $|I|$ copies of the terminal object $1$. The object of sections of $p$, namely $\Pi_! p$ where $!: \Delta I \to 1$ is the unique map and $\Pi_!: E/\Delta I \to E/1 \simeq E$ is right adjoint to the pullback $I^*: E \to E/\Delta I$, is the product $\prod_i X_i$ in $E$. So completeness is certainly a necessary condition for $\mathcal{P}(C)$ to be a topos. 
Much of the study of small cocompletions is bound up with the study of exact completions. Just to cite one result (Lemma 3 in this paper of Rosický): if $C$ is finitely complete, then $\mathcal{P}(C)$ is equivalent to $(\Sigma C)_{ex}$ where $\Sigma C$ is the coproduct completion of $C$ and the subscript denotes exact completion. One expert on properties of exact completions, besides Mike Shulman, is Matías Menni, and so you should probably consult some of his papers for properties of exact completions, starting with his thesis. See also here.