Mike Barr on the question: "is every small topos fully representable into a set-valued functor category?": Unfortunately, a beautiful observation of Makkai's shows that that is impossible. Makkai pointed out that under a full embedding that preserves finite limits, finite sums and epis the boolean algebra of complemented subobjects, which is classified by maps into 1 + 1, would have to be preserved. But in a functor category that lattice is complete and atomic (that is, completely distributive), so that fact, which is not true for toposes in general, becomes a necessary condition for the existence of an embedding. (Is it sufficient?) One can go further. The embedding will necessarily preserve all finite colimits and all infinite coproducts that happen to exist. Extend the Makkai observation as follows. For any pair of objects A and B note that the lattice of partial maps with complemented domains can be constructed as the set of maps from A to 1+B (give 1+B the "flat ordering" of CS: start with B with the trivial partial ordering and adjoin a bottom). The fact that it's a complete lattice is enough to show that arbitrary disjoint unions are coproducts. There is an argument for the case that the topos does not have a natural numbers object but I'll assume here that it does have such. The standard points of the natural numbers object are complemented, hence their union must exist; that union is a coproduct; it must therefore be the entire natural numbers object. It is preserved by the embedding. In Aspects of Topoi there's a lemma entitled "one coequalizer for all" that now suffices to show that the representation preserves all coequalizers.