  Let $\mathbf{C}$ be a Cauchy-complete category. The subtoposes of
$[\mathbf{C}^{\mathrm{op}}, \mathbf{Set}]$ are sometimes all of the form
$[\mathbf{D}^{\mathrm{op}}, \mathbf{Set}]$ where $\mathbf{D}$ is a full
Cauchy-complete subcategory of $\mathbf{C}$. This is the case for instance when
$\mathbf{C}$ is finite, an Artinian poset, or the simplex category. In order to
unify these situations, we give two formulations of a sufficient condition. The
first formulation involves a two-player game, and the second formulation
combines two "local" properties of $\mathbf{C}$.
