Let $X$ be a scheme and $\mathcal S$ a site which is a full subcategory of the category $Aff/X$ of affine schemes with a map to $X$. If I understand correctly, the category $QCoh^\mathcal S(X)$ of $\mathcal S$-quasicoherent sheaves is the global sections of the $\mathcal S$-stackification of the functor $Mod: \mathcal S^{op} \to Cat$, $(Spec A \to X) \mapsto Mod_A$, where $Mod_A$ is the category of $A$-modules.

The interesting phenomenon (cf. the stacks project, which is probably using slightly different definitions) is that for reasonable $\mathcal S$ (the above link gives precise conditions), $QCoh^\mathcal S(X)$ is actually independent of $\mathcal S$. I'm looking for a high-concept explanation of this fact.

The best I can figure is the following. For reasonable topologies, the functor $Mod: \mathcal S \to Cat$ is already a stack, so its global sections can be computed using an $\mathcal S$-cover of the terminal object (which I've technically left out of the category $\mathcal S$, but that's okay). Since $X$ is a scheme, it comes with a Zariski cover, which is also a $\mathcal S$-cover for reasonable topologies. Thus $QCoh^\mathcal S(X)$ is simply computed in the same way for reasonable topologies $\mathcal S$, so of course it agrees.

This suggests that a statement of this meta-principle (an alternative to the one found in the stacks project) would say that

Claim:Let $X$ be a scheme $\mathcal S$ be a full subcategory of $Aff/X$, equipped with a Grothendieck topology. Then $QCoh^\mathcal S(X) = QCoh^{Zariski}(X)$ if

$Mod : \mathcal S^{op} \to Cat$ is a stack, and

Every Zariski cover is an $\mathcal S$-cover.

**Questions:**

Is the above claim correct?

If not (or if so!) is there some other high-concept way to see that $QCoh^\mathcal S(X)$ is independent of $\mathcal S$ for reasonable $\mathcal S$?