  In the previous paper, we constructed a homomorphism from the affine Yangian
associated with $\widehat{\mathfrak{sl}}(n)$ to the standard degreewise
completion of the affine Yangian associated with
$\widehat{\mathfrak{sl}}(n+1)$. By using this homomorphism, we construct a
homomorphism from the affine Yangian to the universal enveloping algebra of a
non-rectangular $W$-algebra, which is different from the one in \cite{U7}. We
also show that this homomorphism is related to the one from the quantum affine
algebra associated with $\widehat{\mathfrak{sl}}(n)$ to the quantum affine
algebra associated with $\widehat{\mathfrak{sl}}(n+1)$ given by Li.
