Pullback commutes with dual for locally free sheaf of finite rank Let $ f:X\rightarrow Y$ be a morphism of ringed spaces. Let $ \mathscr{E} $ be an $\mathcal{O}_Y$ module that is locally free of finite rank. I want to show that $ (f^{*}\mathscr{E})^\vee\cong f^*(\mathscr{E}^\vee)$ where $\vee$ denotes the dual.
I can easily see how stalks are isomorphic since direct limit and tensor product commute with direct sum. What I'm unable to do is construct a morphism from one of these modules to the other. Once I have it, I think I can show that it induces isomorphisms on stalks. Could you please help me?
This is a step in proving the projection formula, so I cannot use this formula here.

$\def\H{{\mathcal{Hom}}}$This follows from this question where it is proved that if $E$ and $F$ are locally free of finite rank then there is a natural isomorphism 
$$
f^*\H_Y(E,F) \to \H_X(f^*E,f^*F). 
$$
In your particular case we have $F = \mathcal{O}_Y$. Then $\H_Y(E,\mathcal{O}_Y) = E^\vee$ and $f^*\mathcal{O}_Y = \mathcal{O}_X$ so we get that
$$
f^*(E^\vee) \cong \H_X(f^*E,\mathcal{O}_X) = (f^*E)^\vee. 
$$
