I was reading Lee's smooth manifold in page 228 when talking about the Lie derivative.
In the Lemma 9.36,the proof of $\mathcal{L}_V W$ is a smooth vector field,it seems only the proof that $\mathcal{L}_VW$ is smooth is presented,the proof that it is a vector field is not shown in the book.
We can prove this since the expression satisfy the product rule:
$$(\mathcal{L}_VW)_p = \lim_{t\to 0} \frac{d(\theta_{-t})_{\theta_t(p)}(W_{\theta_t(p)}) -W_p}{t}$$ acts on the product function $fg$ correct?