Differential forms: difference between $\operatorname{Alt}^nT_pM$ and $\Omega^nM$?

Question

I was wondering whether the following diagram

$\require{AMScd}$ \begin{CD} T_pM @>Df>> T_pN\\ @V \operatorname{Alt}^nDf\circ \omega V V= @VV \omega V\\ \operatorname{Alt}^nT_pM @<<\operatorname{Alt}^nDf< \operatorname{Alt}^nT_pN \end{CD}

where $M,N$ are both smooth manifolds of dimension $n$, $Df$ is the differential (push-forward) of $f\colon M\to N$ and $\operatorname{Alt}^nDf$ is the pull-back of the $n$-form $\omega$ on $N$ is a reasonable to way to "think of" a differential form $\omega$ and the pull-back in general.

I know that $\operatorname{Alt}^n$ is a contravariant functor that acts on linear maps $f\colon V\to W$ between $n$-dimensional vector spaces. However, there is also the contravariant functor $\Omega^n$ that acts on differentiable maps $f\colon M\to N$ between $n$-dimensional manifolds and i was wondering if the following diagram is in fact describing the same scenario as the diagram above:

$\require{AMScd}$ \begin{CD} M @>f>> N\\ @V \Omega^n f\circ \omega_p V V= @VV \omega_p V\\ \Omega^nM @<<\Omega^nf< \Omega^nN \end{CD}

So maybe i failed to be precise in what i mean, so my two questions are:

1. Is $\operatorname{Alt}^n T_pM$ equivalent to $\Omega^n M$?
2. Are the two diagrams actually describing correctly the behaviour of $n$-forms or is any of the diagrams misleading/wrong?

Edit: As pointed out in the comments, i am assuming both diagrams to be point-wise, i.e. with respect to $p \in M$ respectively $f(p) \in N$.