 Dolbeault closedness of Chern curvature

In a holomorphic local frame the Chern matrix is $A=H^{-1}\partial H$. The identity $\partial A+A\wedge A=0$ gives vector-bundle curvature $\Theta =\bar{\partial }A$ of type $(1,1)$. Therefore ${\bar{\partial }}_{\mathrm{End}E}\Theta =0$, allowing the vector-bundle curvature to represent a bundle-valued Dolbeault class.