For an $n$-dimensional manifold $M$, and a smooth vector bundle $E$ over $M$, it is well-known (see Voisin, Huybrechts) that there exists an operator d, built locally from the usual exterior derivative, that acts on $\Gamma^{\infty}(E) \otimes_{C^{\infty}} \Omega^{\bullet}(M)$ so as to give a complex
$$
\Gamma^{\infty}(E) \overset{d}{\to} \Gamma^{\infty}(E) \otimes_{C^{\infty}} \Omega^{1}(M) \overset{d}{\to} \cdots \overset{d}{\to} \Gamma^{\infty}(E) \otimes_{C^{\infty}} \Omega^{n}(M)\overset{d}{\to} 0.
$$

As I prefer global constructions, I began to wonder how one would construct this globally. To apply id $\otimes$ d to $\Gamma^{\infty}(E) \otimes_{C^{\infty}} \Omega^{\bullet}(M)$ is of course not well-defined since we are tensoring over ${C^{\infty}(M)}$. So what can one do . . .?