 Local formula for the Chern connection on a vector bundle

In a holomorphic local frame of a Hermitian holomorphic vector bundle, let $H$ be the matrix of its Hermitian metric on a holomorphic vector bundle. With the convention $\nabla e=eA$, the Chern connection hasThe condition ${\nabla }^{0,1}={\bar{\partial }}_E$ forces $A$ to have type $(1,0)$, while metric compatibility forces this formula, proving both existence and uniqueness.