  We prove that the quantum unipotent coordinate algebra $A_q(\mathfrak{n}(w))\
$ associated with a symmetric Kac-Moody algebra and its Weyl group element $w$
has a monoidal categorification as a quantum cluster algebra. As an application
of our earlier work, we achieve it by showing the existence of a quantum
monoidal seed of $A_q(\mathfrak{n}(w))$ which admits the first-step mutations
in all the directions. As a consequence, we solve the conjecture that any
cluster monomial is a member of the upper global basis up to a power of
$q^{1/2}$.
