Curvature difference of two Chern connections ($F_{D_1}-F_{D_2}=\bar{\partial }a$)

Let $D_1$ and $D_2$ be Chern connections on the same holomorphic vector bundle, possibly for different Hermitian metrics, and put $a=D_1-D_2$. Then $a$ has type $(1,0)$ andIndeed, the curvature difference formula has only types $(2,0)$ and $(1,1)$; both Chern curvatures have type $(1,1)$, so the $(2,0)$ part vanishes and the remaining part is $D_2^{0,1}a={\bar{\partial }}_{\mathrm{End}E}a$.