  We study monoidal categorifications of certain monoidal subcategories
$\mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras,
whose cluster algebra structures coincide and arise from the category of
finite-dimensional modules over quiver Hecke algebra of type A${}_\infty$. In
particular, when the quantum affine algebra is of type A or B, the subcategory
coincides with the monoidal category $\mathcal{C}_{\mathfrak{g}}^0$ introduced
by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster
monomials are real simple modules over quantum affine algebras.
