I'm reading about vector bundles from somme lecture notes written by Matvei Libine. He say that
If $E \rightarrow M $ is a vector bundle, a covariant derivative on $E$ is a differential operator $$\nabla : \Gamma(E,M) \rightarrow \Gamma(M, T^*M \otimes E)$$ wich satisfies the Leibniz's rule; that is, if $s \in \Gamma(E,M)$ and $f \in C^\infty (M)$, then $$\nabla(fs) = df \otimes s + f \nabla s. $$ Note that a covariant derivative extends in a unique way to a map $$\nabla : \Omega^*(M,E) \rightarrow \Omega^{*+1}(M,E)$$ that satisfies the Leibniz's rule: if $\alpha \in \Omega^k(M)$ and $\beta \in \Omega^*(E,M) $, then $$ \nabla(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge \nabla \beta$$.
Could someone please explain to me why and how does $\nabla$ extend in a unique way to a map $\nabla : \Omega^*(M,E) \rightarrow \Omega^{*+1}(M,E)$ that satisfies the Leibniz rule ?