  This paper first investigates solvability of Hermitian-Einstein equation on a
Hermitian holomorphic vector bundle on the complement of an arbitrary closed
subset in a compact complex manifold. The uniqueness of Hermitian-Einstein
metrics on a Zariski open subset in a compact K\"{a}hler manifold was only
figured out by Takuro Mochizuki recently, the second part of this paper gives
an affirmative answer to a question proposed by Takuro Mochizuki and based on
which it leads to an alternative approach to the unique issue. We also prove
stability from solvability of Hermitian-Einstein equation, which together with
the classical existence result of Carlos Simpson in particular establish a
Kobayashi-Hitchin bijective correspondence. The argument is also effective for
uniqueness result and Kobayashi-Hitchin bijective correspondence on
$\mathbb{C}$, as well as on non-K\"{a}hler and semi-stable contexts.
