= Solution
Evaluate the <pullback of a covector> on the coordinate basis vector $\partial_i$. Its pushforward is $(\partial_i y^\alpha)\partial_\alpha$, so
$$
\boxed{(\phi^*\omega)_i
=(\phi^*\omega)(\partial_i)
=\omega(d\phi\,\partial_i)
=\frac{\partial y^\alpha}{\partial x^i}\omega_\alpha.}
$$
Equivalently, pulling back the coordinate one-forms gives $\phi^*(dy^\alpha)=d(y^\alpha\circ\phi)=(\partial_i y^\alpha)dx^i$. \b[The same Jacobian enters the vector pushforward and the covector pullback], with the different index contractions reflecting their duality.
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