Since I got a very good answer to my previous question, 217585, I am asking a sequel by moving up a level.
Let $\mathfrak{g}$ be a simple Lie algebra. Then we have the Yangian $Y(\mathfrak{g})$ and its category of finite dimensional representations, $Y(\mathfrak{g})-\mathrm{mod}$.
We also have the affine Lie algebra $\widehat{\mathfrak{g}}$, its derived algebra $\widehat{\mathfrak{g}}'$ and the quantised enveloping algebra $U_q(\widehat{\mathfrak{g}}')$ and its category of finite dimensional representations, $U_q(\widehat{\mathfrak{g}}')-\mathrm{mod}$.
Then my question is whether $U_q(\widehat{\mathfrak{g}}')-\mathrm{mod}$ can be obtained from $Y(\mathfrak{g})-\mathrm{mod}$ by deformation quantisation? The Yangian, and therefore $Y(\mathfrak{g})-\mathrm{mod}$, is canonically constructed from $\mathfrak{g}$ and a related question is whether this holds for $U_q(\widehat{\mathfrak{g}}')-\mathrm{mod}$?
This looks plausible for the categories as $Y(\mathfrak{g})-\mathrm{mod}$ gives rational solutions of the Yang-Baxter equation and $U_q(\widehat{\mathfrak{g}}')-\mathrm{mod}$ gives trigonometric solutions of the Yang-Baxter equation. However it looks implausible for the Hopf algebras as $Y(\mathfrak{g})$ is based on $\mathfrak{g}[t]$ and $U_q(\widehat{\mathfrak{g}}')$ on $\mathfrak{g}[t,t^{-1}]$.
Note that this question is not answered by my previous question as the tensor product in $Y(\mathfrak{g})-\mathrm{mod}$ is not symmetric.
