Let $(E, \nabla)$ be a vector bundle with connection over a manifold $M$. Denote by $\Omega^k(E)$ the set of $k$-forms valued in $E$: that is, \[ \Omega^k(E) = \Gamma(\Lambda^k T^* M \otimes E) = \Omega^k(M) \otimes \Gamma(E). \] (Note the inconsistency in notation: $M$ is a rank zero vector bundle over $M$, but $\Omega^k(M)$ does not mean the $M$-valued $k$-forms on $M$; it means the standard $k$-forms, and one really ought to write $\Omega^k(\mathbb{R} \times M)$. But this abuse is standard.)