We can define vector fields on manifolds in two ways. The way I first saw was that a vector field was a linear map $C^\infty(M) \to C^\infty(M)$ satisfying the Leibniz rule (aka product rule). We can also define a vector field to be a smooth section of $TM \to M$.
I get that given a section $s$ of $TM \to M$, we can define $\hat{s}(f)(p) = s(p)(f)$, but I don't understand why $\hat{}:Sec(TM \to M) \to Vect(M)$ is an isomorphism. It's clearly linear, but I don't see why it's injective or surjective.
$s \in Sec(TM \to M)$ has a left inverse (the inverse being the projection). Maybe that is used to show $\hat{} :Sec(TM \to M) \to Vect(M)$ is injective?