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I have seen the following statement being used in different papers but never saw a proof:

If $f:X\rightarrow Y$ is a flat morphism between normal varieties and $\mathcal F$ is a reflexive sheaf on $Y$ (i.e. $\mathcal F^{\vee\vee}\cong\mathcal F$). Then the pull-back $f^*\mathcal F$ is a reflexive sheaf on $X$.

Does someone know an easy way to prove this or a paper or book where it is proven?

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See Proposition 1.8 in

R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), 121-176. ZBL0431.14004.

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This is a pretty straightforward statement. The point is that pull-back via a flat morphism commutes with the formation of $\mathscr H\!om$ (essentially because then the pull-back is exact).
So the pull-back of the dual is the dual, etc.

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