The comultiplication of modified quantum affine ${\mathfrak{s}\mathfrak{l}}_n$

Let $\dot{\mathbf{U}}({\widehat{\mathfrak{s}\mathfrak{l}}}_n)$ be the modified quantum affine ${\mathfrak{s}\mathfrak{l}}_n$ and let $\mathbf{U}({\widehat{\mathfrak{s}\mathfrak{l}}}_N{)}^{+}$ be the positive part of quantum affine ${\mathfrak{s}\mathfrak{l}}_N$. Let $\dot{\mathbf{B}}(n)$ be the canonical basis of $\dot{\mathbf{U}}({\widehat{\mathfrak{s}\mathfrak{l}}}_n)$ and let $\mathbf{B}(N{)}^{\mathrm{a}\mathrm{p}}$ be the canonical basis of ${\bf U}(\widehat{\frak{sl}}_N)^+$. It is proved in \cite{FS} that each structure constant for the multiplication with respect to $\dot{\mathbf{B}}(n)$ coincide with a certain structure constant for the multiplication with respect to $\mathbf{B}(N{)}^{\mathrm{a}\mathrm{p}}$ for $n<N$. In this paper we use the theory of affine quantum Schur algebras to prove that the structure constants for the comultiplication with respect to $\dot{\mathbf{B}}(n)$ are determined by the structure constants for the comultiplication with respect to $\mathbf{B}(N{)}^{\mathrm{a}\mathrm{p}}$ for $n<N$. In particular, the positivity property for the comultiplication of $\dot{\mathbf{U}}({\widehat{\mathfrak{s}\mathfrak{l}}}_n)$ follows from the positivity property for the comultiplication of ${\bf U}(\widehat{\frak{sl}}_N)^+$.