The regular qcqs lemma states that if $X$ is a qcqs (quasi-compact quasi-separated) scheme and $f \in \Gamma(X, \mathcal{O}_X)$ then $$\Gamma(X, \mathcal{O}_X)_f \cong \Gamma(D(f), \mathcal{O}_X)$$ where $D(f)$ is the set of points $x$ of $X$ on which $f$ does not vanish (i.e. $f_x \neq 0$).
My question is, does this hold if we replace $\mathcal{O}_X$ with a quasi-coherent sheaf of modules $\mathcal{M}$? That is: $$\Gamma(X, \mathcal{M})_f \cong \Gamma(D(f), \mathcal{M})?$$ I think the steps outlined in Hartshorne Exercise II.2.16 (p.81) should work as long as we find a map $\Gamma(X, \mathcal{M})_f \to \Gamma(D(f), \mathcal{M})$ to get the game started.
For the existance of this map, I would argue as follows. First let $R = \Gamma(X, \mathcal{O}_X)$. We have the restriction map $\Gamma(X, \mathcal{M}) \to \Gamma(D(f), \mathcal{M})$. Localize at $f$ to obtain a map $\Gamma(X, \mathcal{M})_f \to \Gamma(D(f), \mathcal{M})_f$. But now observe that $\Gamma(D(f), \mathcal{M})$ is a $\Gamma(D(f), \mathcal{O}_X) = R_f$-module (where this equality follows by the standard qcqs lemma). It follows that $\Gamma(D(f), \mathcal{M})_f = \Gamma(D(f), \mathcal{M})$.
Does this look correct? Would you know of any reference where the lemma is presented in this form? I am a little nervous because I haven't found a place where it is given in this generality.