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