Two Homomorphisms from the affine Yangian associated with $\widehat{\mathfrak{s}\mathfrak{l}}(n)$ to the affine Yangian associated with $\widehat{\mathfrak{s}\mathfrak{l}}(n+1)$

We construct a homomorphism from the affine Yangian $Y_{\mathrm{\hslash },\epsilon +\mathrm{\hslash }}(\widehat{\mathfrak{s}\mathfrak{l}}(n))$ to the affine Yangian $Y_{\mathrm{\hslash },\epsilon }(\widehat{\mathfrak{s}\mathfrak{l}}(n+1))$ which is different from the one in arXiv:2312.09933. By using this homomorphism, we give a homomorphism from $Y_{\mathrm{\hslash },\epsilon }(\widehat{\mathfrak{s}\mathfrak{l}}(n))\otimes Y_{\mathrm{\hslash },\epsilon +n\mathrm{\hslash }}(\widehat{\mathfrak{s}\mathfrak{l}}(m))$ to $Y_{\mathrm{\hslash },\epsilon }(\widehat{\mathfrak{s}\mathfrak{l}}(m+n))$. As an application, we construct a homomorphism from the affine Yangian $Y_{\mathrm{\hslash },\epsilon +n\mathrm{\hslash }}(\widehat{\mathfrak{s}\mathfrak{l}}(m))$ to the centralizer algebra of the pair of affine Lie algebras $(\widehat{\mathfrak{g}\mathfrak{l}}(m+n),\widehat{\mathfrak{g}\mathfrak{l}}(n))$ and the coset vertex algebra of the pair of rectangular $W$-algebras $({\mathcal{W}}^k(\mathfrak{g}\mathfrak{l}(2m+2n),(2^{m+n})),{\mathcal{W}}^{k+m}(\mathfrak{s}\mathfrak{l}(2n),(2^n)))$.