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We construct an exact tensor functor from the category $\mathcal{A}$ offinite-dimensional graded modules over the quiver Hecke algebra of type$A_\infty$ to the category $\mathscr C_{B^{(1)}_n}$ of finite-dimensionalintegrable modules over the quantum affine algebra of type $B^{(1)}_n$. Itfactors through the category $\mathcal T_{2n}$, which is a localization of$\mathcal{A}$. As a result, this functor induces a ring isomorphism from theGrothendieck ring of $\mathcal T_{2n}$ (ignoring the gradings) to theGrothendieck ring of a subcategory $\mathscr C^{0}_{B^{(1)}_n}$ of $\mathscrC_{B^{(1)}_n}$. Moreover, it induces a bijection between the classes of simpleobjects. Because the category $\mathcal T_{2n}$ is related to categories$\mathscr C^{0}_{A^{(t)}_{2n-1}}$ $(t=1,2)$ of the quantum affine algebras oftype $A^{(t)}_{2n-1}$, we obtain an interesting connection between thosecategories of modules over quantum affine algebras of type $A$ and type $B$.Namely, for each $t =1,2$, there exists an isomorphism between the Grothendieckring of $\mathscr C^{0}_{A^{(t)}_{2n-1}}$ and the Grothendieck ring of$\mathscr C^{0}_{B^{(1)}_n}$, which induces a bijection between the classes ofsimple modules.

Monoidal categories of modules over quantum affine algebras of type A and B