Consider a topos $\mathcal E$. Butz and Moerdijk, in Representing topoi by topological groupoids, par. 2, say that one can find an object $S\in \mathcal E$ such that the subobjects of its powers (i.e., sheaves $B\subset S^n$ for some $n$) generate $\mathcal E$. They suggest to use $S=\sum_{c\in C}c$, where $C$ is a site of definition of $\mathcal E$. (I suppose they identify $c$ with $Yon(c)$.)
I have difficulties in finding out why this works. Perhaps my problem is that I don't understand in what sense the word "generate" is used.
Thank you in advance.
Kevin Carlson's and Pece's answers are correct in explaining when a family of objects is generating. In every Grothendieck topos the collection of representable sheaves (i.e., the sheafifications of the Yoneda embedding are generating (this follows from sheafification applied to the contents of the Yoneda lemma). In general, if a site is at hand then the collection of subobjects of the co-product over all objects of that site has the property that its subobjects generate the topos. However, most often, much smaller objects are sufficient. If, for example, the topos is generated by a syntatic site, then the co-product over all finite products of finite types suffces.