Things are algebraic over $\mathbb{C}$ (or maybe even complex analytic.)
Suppose we have a short exact sequence of torsion free coherent sheaves on a surface $X$ (normal maybe?):
$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0.$
Does it induce:
$0\rightarrow A^{\vee\vee}\rightarrow B^{\vee\vee}\rightarrow C^{\vee\vee}\rightarrow 0?$