= Solution
A type $(0,s)$ <covariant tensor> is a multilinear form on $s$ vectors. For arbitrary $V_1,\ldots,V_s\in T_p\mathcal M$, the <pullback of a covariant tensor> is
$$
(\phi^*\omega)(V_1,\ldots,V_s)
=\omega(d\phi V_1,\ldots,d\phi V_s).
$$
Projection of all covariant slots means $(P\omega)(X_1,\ldots,X_s)=\omega(PX_1,\ldots,PX_s)$. Since every pushed-forward vector is already tangential,
$$
\begin{aligned}
[\phi^*(P\omega)](V_1,\ldots,V_s)
&=\omega(Pd\phi V_1,\ldots,Pd\phi V_s)\\
&=\omega(d\phi V_1,\ldots,d\phi V_s).
\end{aligned}
$$
Equality on all arguments proves \b[$\phi^*\omega=\phi^*(P\omega)$]. No antisymmetry is needed: this holds for every <covariant tensor>, not only for <differential forms>.
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