Notes on a homomorphism from the affine Yangian associated with $\widehat{\mathfrak{s}\mathfrak{l}}(n)$ to the affine Yangian associated with $\widehat{\mathfrak{s}\mathfrak{l}}(n+1)$

In the previous paper, we constructed a homomorphism from the affine Yangian associated with $\widehat{\mathfrak{s}\mathfrak{l}}(n)$ to the standard degreewise completion of the affine Yangian associated with $\widehat{\mathfrak{s}\mathfrak{l}}(n+1)$. In this article, by using this homomorphism, we construct homomorphisms from the affine Yangian to the universal enveloping algebra of a non-rectangular $W$-algebra of type $A$, which are different from the one in \cite{U7}. Based on this result, we discuss a new definition of the affine shifted Yangian, from which it is expected that there exists a homomorphism to the universal enveloping algebra of a non-rectangular $W$-algebra of type $A$. In the appendix, we also show that this homomorphism is related to the one from the quantum affine algebra associated with $\widehat{\mathfrak{s}\mathfrak{l}}(n)$ to the quantum affine algebra associated with $\widehat{\mathfrak{s}\mathfrak{l}}(n+1)$ given by Li.