Thanks Manuel, I'll check this out :)

Is the following true, and if so, is there a proof somewhere?

Let $C$ be a small $\infty$-category closed under finite coproducts, and $X\colon C^{\mathrm{op}}\to \mathcal{S}$ a presheaf on $C$. Then does $X$ preserve finite products iff $(C/X):=C_{/X}\times_C\mathrm{PSh}(C)$ is sifted?

@CharlesRezk I think that's true. Actually, I think C/X should admit finite coproducts (which implies it's sifted). dunno a reference...

Aha. I can't find it in 5.5.8.