Let $X,Y$ be smooth algebraic varieties over $k$, and $f:X\to Y$ a morphism of varieties. Denote by $\mathcal{T}_Y$ and $\Omega_Y$ the tangent and cotangent sheaves respectively.
Why are $f^*\Omega_Y$ and $f^*\mathcal{T}_Y$ dual? Is it true in general that for dual locally free sheaves $\mathcal{F},\mathcal{G}$ that $f^*\mathcal{F}$ and $f^*\mathcal{G}$ are dual?
Sorry if this is obvious, but I can't seem to show this yet.