  This paper concerns the relation between the quantum toroidal algebras and
the affine Yangians of $\mathfrak{sl}_n$, denoted by
$\mathcal{U}^{(n)}_{q_1,q_2,q_3}$ and $\mathcal{Y}^{(n)}_{h_1,h_2,h_3}$,
respectively. Our motivation arises from the milestone work of Gautam and
Toledano Laredo, where a similar relation between the quantum loop algebra
$U_q(L \mathfrak{g})$ and the Yangian $Y_h(\mathfrak{g})$ has been established
by constructing an isomorphism of $\mathbb{C}[[\hbar]]$-algebras
$\Phi:\widehat{U}_{\exp(\hbar)}(L\mathfrak{g})\to
\widehat{Y}_\hbar(\mathfrak{g})$ (with $\ \widehat{}\ $ standing for the
appropriate completions). These two completions model the behavior of the
algebras in the formal neighborhood of $h=0$. The same construction can be
applied to the toroidal setting with $q_i=\exp(\hbar_i)$ for $i=1,2,3$.
  In the current paper, we are interested in the more general relation:
$\mathrm{q}_1=\omega_{mn}e^{h_1/m}, \mathrm{q}_2=e^{h_2/m},
\mathrm{q}_3=\omega_{mn}^{-1}e^{h_3/m}$, where $m,n\in \mathbb{N}$ and
$\omega_{mn}$ is an $mn$-th root of $1$. Assuming $\omega_{mn}^m$ is a
primitive $n$-th root of unity, we construct a homomorphism
$\Phi^{\omega_{mn}}_{m,n}$ from the completion of the formal version of
$\mathcal{U}^{(m)}_{\mathrm{q}_1,\mathrm{q}_2,\mathrm{q}_3}$ to the completion
of the formal version of $\mathcal{Y}^{(mn)}_{h_1/mn,h_2/mn,h_3/mn}$. We
propose two proofs of this result: (1) by constructing the compatible
isomorphism between the faithful representations of the algebras; (2) by
combining the direct verification of Gautam and Toledano Laredo for the
classical setting with the shuffle approach.
