  Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine
$ADE$ type, and $\mathcal{C}_{\mathfrak{g}}^0$ the Hernandez-Leclerc category
of finite-dimensional $U_q'(\mathfrak{g})$-modules. For a suitable infinite
sequence $\widehat{w}_0= \cdots s_{i_{-1}}s_{i_0}s_{i_1} \cdots$ of simple
reflections, we introduce subcategories $\mathcal{C}_{\mathfrak{g}}^{[a,b]}$ of
$\mathcal{C}_{\mathfrak{g}}^0$ for all $a \le b \in \mathbb{Z}\sqcup\{ \pm
\infty \}$. Associated with a certain chain $\mathfrak{C}$ of intervals in
$[a,b]$, we construct a real simple commuting family $M(\mathfrak{C})$ in
$\mathcal{C}_{\mathfrak{g}}^{[a,b]}$, which consists of Kirillov-Reshetikhin
modules. The category $\mathcal{C}_{\mathfrak{g}}^{[a,b]}$ provides a monoidal
categorification of the cluster algebra
$K(\mathcal{C}_{\mathfrak{g}}^{[a,b]})$, whose set of initial cluster variables
is $[M(\mathfrak{C})]$. In particular, this result gives an affirmative answer
to the monoidal categorification conjecture on $\mathcal{C}_{\mathfrak{g}}^-$
by Hernandez-Leclerc since it is $\mathcal{C}_{\mathfrak{g}}^{[-\infty,0]}$,
and is also applicable to $\mathcal{C}_{\mathfrak{g}}^0$ since it is
$\mathcal{C}_{\mathfrak{g}}^{[-\infty,\infty]}$.
