A well-known theorem states that a category **E** is a Grothendieck topos if and only if it can be embedded as a reflective subcategory of a presheaf category whose reflector preserves finite limits.

In their paper *Grothendieck quasitoposes*, Garner and Lack proved that a category **E** is a Grothendieck quasitopos (the category of sheaves for one topology that are separated for another topology) if and only if it can be embedded as a reflective subcategory of a presheaf category whose reflector has stable units (i.e. preserves all pullbacks over objects in the subcategory) and preserves monomorphisms.

Is there a characterization of when a category **E** can be embedded as a reflective subcategory of a presheaf category whose reflector preserves finite products? (The reflector preserving finite products is equivalent to the reflective subcategory being an *exponential ideal*, i.e. $Y^X$ lies in the subcategory as soon as $Y$ does.) Of course, any such category must be locally presentable and cartesian closed. Can this be extended to a set of conditions which is both necessary and sufficient?