Is "quasi-coherent" an fpqc-local property of modules? 
Let $f : X \to Y$ be an fpqc morphism of schemes, and let $\mathcal{G}$ be an $\mathcal{O}_{Y}$-module (on the small Zariski site) such that $f^{\ast}\mathcal{G}$ is quasi-coherent. Is $\mathcal{G}$ necessarily quasi-coherent?

I'd be happy to see any answers to the above question with "fpqc" replaced by "fppf" or "etale" as well.
 A: Yes, this is an application of descent theory (with a bit of care).  Let $F = f^{\ast}(G)$ on $X$, a quasi-coherent $O_X$-module by hypothesis.  Then for the maps $p_1, p_2: X \times_Y X \rightrightarrows X$ we have an evident composite isomorphism $$\theta: p_1^{\ast}(F) \simeq (f \circ p_1)^{\ast}(G) = (f \circ p_2)^{\ast}(G) \simeq p_2^{\ast}(F)$$ that satisfies the usual cocycle condition; i.e., $\theta$ is a descent datum. Hence, by fpqc descent for quasi-coherent sheaves, we obtain a quasi-coherent $O_Y$-module $G'$ and an $O_X$-linear isomorphism $\alpha: f^{\ast}(G') \simeq F := f^{\ast}(G)$ respecting the descent data on both sides.
Let $f':X \times_Y X \rightarrow Y$ be the natural map, and $F' = {f'}^{\ast}(G)$, so $F'$ is naturally identified with each of $p_1^{\ast}(F)$ and $p_2^{\ast}(F)$ compatibly with $\theta$ (via how $\theta$ is defined).  The equality of $f'$ with $f \circ p_1$ and $f \circ p_2$ thereby defines two $O_Y$-linear maps
$$f_{\ast}(F) \rightrightarrows f'_{\ast}(F')$$
whose equality is the quasi-coherent $G'$ by design.  But $G$ is visibly an $O_Y$-submodule of $f_{\ast}(F)$ (since $f$ is surjective and faithfully flat between local rings) and as such is contained inside the equalizer $G'$, so we have $G \subset G'$. The problem of checking equality thereby reduces to comparing stalks at each $y \in Y$.  
If $x \in f^{-1}(y)$ is a point then $O_y \rightarrow O_x$ is faithfully flat, so by fpqc descent for modules we see that
$$G_y = \ker(F_x \rightrightarrows F_x \otimes_{O_y} O_x).$$
But the right side coincides with $G'_y$ since $G'$ is a quasi-coherent descent of $F$, so $G_y = G'_y$ via the natural map; i.e., the inclusion $G \subset G'$ is an equality on $y$-stalks.
