We study monoidal categorifications of certain monoidal subcategories scriptCJ$\mathcal {C}_J$ of finite‐dimensional modules over quantum affine algebras, whose cluster algebra structures on their Grothendieck rings Kfalse(CJfalse)$K(\mathcal {C}_J)$ are closely related to the category of finite‐dimensional modules over quiver Hecke algebra of type A∞$A_\infty$ via the generalized quantum Schur–Weyl duality functors. In particular, when the quantum affine algebra is of type A$A$ or B$B$, the subcategory coincides with the monoidal category scriptCfrakturg0$\mathcal {C}_{\mathfrak {g}}^0$ introduced by Hernandez–Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.