Equivalence between module categories over quiver Hecke algebras and Hernandez–Leclerc's categories in general types

We prove in full generality that the generalized quantum affine Schur–Weyl duality functor, introduced by Kang–Kashiwara–Kim, gives an equivalence between category of finite-dimensional modules over a quiver Hecke algebra and certain subcategory which is generalization Hernandez–Leclerc's CQ. This was previously proved untwisted ADE types Fujita using geometry varieties, not applicable general. Our proof purely algebraic, so can be extended uniformly to general types.