For a Dynkin quiver
Q
Q
(of type
A
D
E
\mathrm {ADE}
), we consider a central completion of the convolution algebra of the equivariant
K
K
-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc’s monoidal category
C
Q
\mathcal {C}_{Q}
of modules over the quantum loop algebra
U
q
(
L
g
)
U_{q}(L\mathfrak {g})
via Nakajima’s homomorphism. As an application, we show that Kang-Kashiwara-Kim’s generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with
Q
Q
and Hernandez-Leclerc’s category
C
Q
\mathcal {C}_{Q}
, assuming the simpleness of some poles of normalized
R
R
-matrices for type
E
\mathrm {E}
.