TITLE: Taking the category of sheaves is symmetric monoidal
QUESTION [10 upvotes]: Let $M$ and $N$ be topological spaces.
Let $\operatorname{Sh}(M)$ denote the presentable $\infty$-category of space-valued sheaves on $M$.
It seems to me that the equivalence
$$\operatorname{Sh}(M) \otimes \operatorname{Sh}(N) = \operatorname{Sh}(M \times N)$$
where $\otimes$ is the standard symmetric monoidal structure of $\operatorname{Pr^L}$, the $\infty$-category of presentable categories with left adjoints, is well-known.
However, I cannot find a proof in the literature.
My question is if there is a standard reference for this statement? Or if there is a "simple proof" of it by assuming some well-known results?
By the way, I believe that I was once told that it's in Lurie but I've failed to find it in either Higher Topos Theory or Higher Algebra.

REPLY [12 votes]: Provided at least one of $M$ and $N$ is locally compact, the $\infty$-topos $\mathrm{Sh}(M \times N)$ is the product of $\mathrm{Sh}(M)$ and $\mathrm{Sh}(N)$ in $\mathrm{RTop}$. This is HTT 7.3.1.11.

Products in $\mathrm{RTop}$ can be computed as tensor products in $\mathrm{Pr^L}$. This is HA Example 4.8.1.19.


HA doesn't include a complete proof of the latter, but the case where one of the factors is of the form $\mathrm{Sh}(M)$ is essentially HTT 7.3.3.9, up to the matter of identifying $\mathrm{Sh}(M; \mathrm{Sh}(N))$ with the tensor product $\mathrm{Sh}(M) \otimes \mathrm{Sh}(N)$, for which HA 4.8.1.17 should be useful.