||We construct an exact tensor functor from the category A of finite-dimensional graded modules over the quiver Hecke algebra of type A(infinity) to the category L-Bn(1) of finite-dimensional integrable modules over the quantum affine algebra of type B-n((1)). It factors through the category T-2n, which is a localization of A. As a result, this functor induces a ring isomorphism from the Grothendieck ring of T-2n (ignoring the gradings) to the Grothendieck ring of a subcategory L-Bn(1)(0) of L-Bn(1). Moreover, it induces a bijection between the classes of simple objects. Because the category T-2n is related to categories L-A2n-1(t)(0) (t=1,2) of the quantum affine algebras of type A(2n-1)((t)), we obtain an interesting connection between those categories of modules over quantum affine algebras of type A and type B. Namely, for each t=1,2, there exists an isomorphism between the Grothendieck ring of l(A2n-1(t))(0) and the Grothendieck ring of l(Bn(1))(0), which induces a bijection between the classes of simple modules.