Tensor product of sites

Question

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).

1. Is it true that this category $H$ is a Grothendieck topos? (I have almost no doubt about this)
2. 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.

Answer 1

The category $H$ can be described as the category of $E$-valued sheaves on $D$, or $F$-valued sheaves on $C$.

You get a site by taking the category $C \times D$ and taking the topology generated by the $(c_i,d) \to (c,d)$ for $c_i \to c$ a covering family in $C$, and the $(c,d_i) \to (c,d)$ for $d_i \to d$ a covering family in $D$.

(or equivalently, the generators are the $\{(c_i,d_j) \to (c,d)\}_{i\in I,j \in J}$ for $c_i \to c$ and $d_j \to d$ covering families in $C$ and $D$ respectively - this is probably easier to check the claim above with this definition).

In particular, $H$ is a Grothendieck topos.

In terms of the construction of a semi-direct product site in the Elephant, this should corresponds to the case of a "constant" internal site.

Finally, this construction makes $H$ the product of $E$ and $F$ in the category of Grothendieck topos and geometric morphisms (in the geometric direction, so coproduct in the algebraic direction).