I am reading Introduction to Symplectic Topology, by McDuff et al.
At the end of page 97, it seems to suggest that if $\omega$ is a $k$-form over $M$ and $\psi: M \rightarrow M$ is an $\omega$ invariant diffeomorphism. That is, $\psi^* \omega = \omega$, then, $$\iota(\psi^* X) \omega = \psi^*(\iota(X) \omega).$$
Here, $\iota$ represents the inner derivation.
How can I show this?
PS: Actually, in the book, $\psi_t: M \rightarrow M$ is a smooth family of $\omega$ invariant diffeomorphisms. But I guess it is not important for this post.