[All equation numbers reference Wald, Robert M., General relativity, Chicago-London: The University of Chicago Press. XIII, 491 p. 34.50 (1984).]
Consider a subset $O$ of an $n$-dimensional, $C^{\infty}$, real manifold $M$. Let $\psi:O\to V\subset\mathbb{R}^n$ be a chart and let $x^\mu:\mathbb{R}^n\to\mathbb{R}$ denote coordinates corresponding to some choice of basis $\{e_\mu\}$ on $\mathbb{R}^n$. Composition yields the smooth maps $x^\mu\circ \psi:O\to\mathbb{R}$.
Now, the corresponding coordinate components $v^\mu$ of any tangent vector $v\in V_p$ at $p\in O$ are given by
\begin{align}
v^\mu=v(x^\mu\circ\psi(p)), \tag{2.2.6}\label{v mu}
\end{align}
with $v(f)$ viewed as a linear map $v:C^{\infty}(M)\to\mathbb{R}$.
On the other hand, the push-forward map $\psi^*:V_p\to V_{\psi(p)}$ associated with $\psi$ is defined via \begin{align} (\psi^*v)(f)\equiv v(f\circ\psi) \tag{C.1.1}\label{push v}, \end{align} for all smooth $f:\mathbb{R}^n\to\mathbb{R}$. So, it would seem the coordinate components of $\psi^*v$ at $\psi(p)$ are given by the same expression as the coordinate components of $v$ at $p$:
\begin{align} (\psi^*v)^\mu = (\psi^*v)(x^\mu) \overset{\eqref{push v}}{=} v(x^\mu\circ\psi(p)) \overset{\eqref{v mu}}{=}v^\mu \label{*}\tag{$\star$} \end{align}
Likewise, in the basis dual to \begin{align}X_\mu(f)\equiv \left.\frac{\partial}{\partial x^\mu}(f\circ \psi^{-1})\right|_{\psi(p)} \tag{2.2.1} \label{X mu}, \end{align} the coordinate components of any dual vector $w\in {V}^*_p$ are given by \begin{align} w_\mu=w(X_\mu), \tag{c.f. Ch2 Problem 6.b} \label{w mu} \end{align} Since $\psi$ is bijective, its inverse ${\psi^{-1}}:V\to O$ can provide a push-forward $(\psi^{-1})^*:{V}^*_p \to {V}^*_{\psi(p)}$ for cotangent vectors by requiring that for every $v\in V_{\psi (p)}$, \begin{align} ((\psi^{-1})^*w)(v)=w((\psi^{-1})^*v) \tag{c.f. C.1.2} \label{push w} \end{align} The coordinate components of $(\psi^{-1})^*w$ at $\psi(p)$ are given by, \begin{align} ((\psi^{-1})^*w)_\mu = ((\psi^{-1})^*w)\left(\frac{\partial}{\partial x^\mu} \right) \overset{\eqref{push w}}{=} w\left((\psi^{-1})^*\frac{\partial}{\partial x^\mu}\right)\overset{\eqref{X mu}}{=}w(X_\mu)\overset{\eqref{w mu}}{=}w_\mu. \label{**}\tag{$\star\star$} \end{align}
Since tensors at $p$ are defined as multilinear maps over $V_p$ and ${V}^*_p$, one should straightforwardly obtain analogous formulas for tensors of any type using $\eqref{*}$ and $\eqref{**}$.
Question(s): Are formulas \eqref{*} and \eqref{**} and their derivations correct? Is it valid to view these formulas as saying: "The coordinate components (associated with chart $\psi$) of a tensor at $p\in M$ are equal the coordinate components of the tensor's push-forward via $\psi$ to $\psi(p)\in\mathbb{R}^n$" ?