Let $X$ be a noetherian scheme and $(F_n)_{\in \mathbb N}$ an inverse system (indexed by the natural numbers) of locally sheaves of finite rank on $X$ (the ranks may vary). The transition maps $F_{n+1} \rightarrow F_n$ shall be surjective for all $n$.
Then is there a canonical isomorphism
$\mathcal Hom_{\mathcal O_X} (\varprojlim F_n, \mathcal O_X) \simeq \varinjlim \mathcal Hom_{\mathcal O_X}(F_n,\mathcal O_X)$?
Here always Sheaf-Homs are meant, and the limits are at first considered in the category of quasi-coherent modules (where they exist in any case).