I am looking for a reference which shows that the following statements are equivalent for a complex vector bundle $E$:

- $E$ is a holomorphic vector bundle.
- There is a Dolbeault operator $\bar{\partial}_E$, i.e. a $\mathbb{C}$ linear operator $\bar{\partial}_E : \Omega^{0,0}(E) \to \Omega^{0,1}(E)$ which satisfies the Leibniz rule and $\bar{\partial}_E^2 = 0$.

This is stated without proof in Huybrechts' *Complex Geometry: An Introduction*.