(I will use the terminology in the book {\em Sheaves in Geometry and Logic} by Mac Lane and Moerdijk.)
A collection ${\{ G_{\xi} \}}$ of objects in a category $\mathbf{A}$ is said to {\em generate} $\mathbf{A}$ iff any parallel pair of morphisms ${f, g : A \rightarrow B}$ in $\mathbf{A}$ one has ${f = g}$ iff ${f t = g t}$ for all maps ${t : G_{\xi} \rightarrow A}$ with $G_{\xi}$ any object from the collection ${\{ G_{\xi} \}}$.
If $\mathbf{A}$ is a small category then any subcategory ${\mathbf{B} \rightarrow \mathbf{A}}$ induces a subtopos ${\widehat{\mathbf{B}} \rightarrow \widehat{\mathbf{A}}}$ between presheaf toposes.
Is there a characterization of the subtoposes ${\widehat{\mathbf{B}} \rightarrow \widehat{\mathbf{A}}}$ such that ${\mathbf{B} \rightarrow \mathbf{A}}$ is generating (in the sense that the objects in $\mathbf{B}$ form a generating collection)?
I am looking for a characterization that makes sense for subtoposes of elementary toposes. For example, I don't think that ${\mathbf{B}}$ is generating iff the subtopos is dense.