In a proof of a proposition on p.174 of Andrew McInerney's "First Steps in Differential Geometry: Riemannian, Contact, Symplectic", the author derived from $(\phi^*_t \partial_i)$ to $(\phi_{-t})_* \partial_i$ where $\phi_t$ represents the flow generated by a vector field and $\partial_i$ represents a member of the standard basis of the vector field. Here the author claimed that he was using Definition 4.6.12 of the pullback of a tensor field and the fact that $\phi_{-t}$ is the inverse of $\phi_t$.
But according to Definition 4.6.12 (on p. 167 of the book), if $\phi$ is a diffeomorphism, $S$ is a (1,1)-tensor field, $\alpha$ is a one-form and $X$ is a vector field, then
$(\phi^*S)(\alpha, X) = S((\phi^{-1})^*\alpha, \phi_*X)$
Here I don't understand why the author used $\phi_{-t}$ in the derivation as $\partial_i$ is a vector field rather than a one-form.