Two approaches to the definition of a connection

Question

A connection can be defined as a map $$D:\Gamma(E)\to\Gamma(E)\otimes\Gamma(T^*M)$$ satisfying the usual conditions or, equivalently, $$F:\Gamma(TM)\otimes \Gamma(E)\to \Gamma(E).$$

My question is simple: how can I see the equivalence of these two definitions?

Answer 1

Let $s \in \Gamma(E)$ and $V \in \Gamma(TM)$, then the two notions are related by the equality $$D(s)(V) = F(V, s)$$ as elements in $\Gamma(E)$.