topos defined by sets Let $C=Sets$ be the category/site of sets, equipped with the topology defined by surjective families. Why is the associated topos $T$ equivalent to the punctual topos $Sh(pt)\simeq Sets$? (This is claimed in Luc Illusie's article "What is a topos?"). I reason that sheaves on the site $C$ are all presheaves (of sets) on $C$, and so the topos should be the presheaf category $Fun(Sets^{op}, Sets)$. But Yoneda gives an (non-essentially surjective, I believe) embedding of $Sets$ into $Fun(Sets^{op}, Sets)$. Obviously I'm making an error somewhere.

Let $\mathcal{E}$ be a Grothendieck topos. It is a standard fact that the category of _sheaves_ on $\mathcal{E}$ with respect to the canonical topology on $\mathcal{E}$ is equivalent to $\mathcal{E}$ (via the Yoneda embedding): see here. (The canonical topology on $\mathcal{E}$ has as its covering families all jointly epimorphic families.)

The claim in question is the special case where $\mathcal{E} = \mathbf{Set}$.