= Solution
For a <smooth map between manifolds> $\phi:\mathcal M\to\mathcal N$, the four constructions go in the directions dictated by composition and the <differential of a smooth map>.
The <pullback of a smooth function> $f\in C^\infty(\mathcal N)$ is \b[$\phi^*f=f\circ\phi$], a function on $\mathcal M$. The <pushforward of a curve> $\lambda:I\to\mathcal M$ is \b[$\phi_*\lambda=\phi\circ\lambda$], a curve in $\mathcal N$ with the same parameter interval.
For a <tangent vector> $V\in T_p\mathcal M$, its pushforward is \b[$\phi_*V=d\phi_p(V)\in T_{\phi(p)}\mathcal N$]. Intrinsically, treating a <tangent vector> as a derivation on functions,
$$
(\phi_*V)[f]=V[f\circ\phi].
$$
For a <covector> $\eta\in T_{\phi(p)}^*\mathcal N$, the <pullback of a covector> is the dual linear map:
$$
\boxed{(\phi^*\eta)_p(V)=\eta_{\phi(p)}(d\phi_pV).}
$$
No inverse map is required. A pushed-forward field for a general map is a field along that map; it need not assign a unique vector to an image point with several preimages.
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