  We construct an equivalence of graded Abelian categories from a category of
representations of the quiver-Hecke algebra of type $A_1^{(1)}$ to the category
of equivariant perverse coherent sheaves on the nilpotent cone of type $A$. We
prove that this equivalence is weakly monoidal. This gives a
representation-theoretic categorification of the preprojective K-theoretic Hall
algebra considered by Schiffmann-Vasserot. Using this categorification, we
compare the monoidal categorification of the quantum open unipotent cells of
type $A_1^{(1)}$ given by Kang-Kashiwara-Kim-Oh-Park in terms of quiver-Hecke
algebras with the one given by Cautis-Williams in terms of equivariant perverse
coherent sheaves on the affine Grassmannians.
