Suppose $\mathscr{M}$ and $\mathscr{N}$ are locally free sheaves of rank $n$ on the Zariski site of some variety $X_{zar}$. I would like to show $\mathscr{M}\cong \mathscr{N}$ if and only if $\mathscr{M}^{fl}\cong \mathscr{N}^{fl}$ where we take the associated sheaf on the small flat site.
I am confused on how to prove this. At the very least, I can prove the statement in the case of $X$ affine first. The $\Rightarrow$ direction is obvious. My issue is the $\Leftarrow$ direction. I know $\mathscr{M}^{fl}$ (resp. $\mathscr{N}^{fl}$) can be obtained from sheaves on $X_{zar}$ using descent data. If I coulde somehow show that the descent datum uniquely determines $\mathscr{M}$ and $\mathscr{N}$, then I get the isomorphism.
I am reading Milne's Lectures on Etale Cohomology if that helps (this is on p. 79) and they omit the proof of this.
Can such a claim be said of other sites? For instance, with $\mathscr{M}^{et}$ instead. My thought is no since faithfully flat descent seems to be a crucial tool.
Let me just get this off the unanswered this. This is essentially true for any reasonable site. See this for a reference in, what I would imagine, is sufficiently large generality for you.