= Solution
In a <holomorphic local frame> $e$, let $H=(h(e_i,e_j))$ and write $De=eA$. Holomorphic compatibility forces $A^{0,1}=0$, while <metric compatibility> forces the <local formula for the Chern connection on a vector bundle>
$$
\boxed{A=H^{-1}\partial H.}
$$
This proves uniqueness. The formula transforms by the <connection one-form> law under a holomorphic frame change, so the local definitions glue and satisfy both conditions. This proves existence of the unique <Chern connection>.
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