Some textbooks of differential geometry (such as A. McInerney's First Steps in Differential Geometry: Riemannian, Contact, Symplectic) define the pullback of a tensor field with respect to a diffeomorphism. Let $\psi: M \rightarrow N$ be a diffeomorphism with inverse $\psi^{-1}: N \rightarrow M$, and $T$ be an $(r, s)$-tensor field on $N$. Then the pullback of $T$ is defined to be a tensor field on $M$ given by:
$\psi^*T(\alpha_1, \ldots, \alpha_r, v_1, \ldots, v_s) = T(\psi^{-1*}\alpha_1, \ldots, \psi^{-1*}\alpha_r, \psi_*v_1, \ldots, \psi_*v_s)$
where $\alpha_1$, ..., $\alpha_r$ are smooth 1-forms on $M$ and $v_1$, ..., $v_s$ are smooth vector fields on $M$.
I wonder whether the above definition can be extended to pushforward of a tensor field. Let $S$ be a tensor field on $M$. Since we assume that $\psi^{-1}$ exists, isn't it reasonable to define the pushforward of $S$ to be a tensor field on $N$ given by
$\psi_*S(\beta_1, \ldots, \beta_r, w_1, \ldots, w_s) = S(\psi^*\beta_1, \ldots, \psi^*\beta_r, \psi^{-1}_*w_1, \ldots, \psi^{-1}_*w_s)$
where $\beta_1$, ..., $\beta_r$ are smooth 1-forms on $N$ and $w_1$, ..., $w_s$ are smooth vector fields on $N$?
If so, then isn't it possible to define the pushforward of a 1-form with respect to a diffeomorphism since 1-forms are a special type of $(0, 1)$-tensor fields?