Let $C, D$ two Grothendieck sites. Since the corresponding toposes $E, F$ are locally presentable categories, then (by Gabriel-Ulmer duality) they correspond to limit theories $X, Y$ (that is, small categories having all $\lambda$-small limits; here $\lambda$ is chosen larger of two). Now we can consider $X$-objects in $F$ or $Y$-objects in $E$. This gives the same locally-representable category $H$ (this is known as: the tensor product of limit theories is commutative).

- Is it true that this category $H$ is a Grothendieck topos? (I have almost no doubt about this)
- Is there a natural construction (in terms of sites) defining a site $C \otimes D$, the topos of sheaves over which is equivalent to $H$? It is natural to expect that as a category it would be $C \times D$ (then, in view of the currying, this could exactly be treated as $F$-valued sheaves on $C$ or $E$-valued sheaves on $D$). Is it possible to take just the direct product of the covering families in $C$ and $D$ as covering families?

Jonstone (in the $C$ part) defines a semidirect product of sites, but it talks about sites internal to the toposes. I'm not sure if this is related to my question yet.