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Let   $\textsf {X}$   and   $\textsf {X}^{!}$   be a pair of symplectic varieties dual with respect to 3D mirror symmetry. The   $K$  -theoretic limit of the elliptic duality interface is an equivariant   $K$  -theory class   $\mathfrak {m} \in K(\textsf {X}\times \textsf {X}^{!})$  . We show that this class provides correspondences   $$ \begin{align*} &amp; \Phi_{\mathfrak{m}}: K(\textsf{X}) \leftrightarrows K(\textsf{X}^{!}) \end{align*}$$  mapping the   $K$  -theoretic stable envelopes to the   $K$  -theoretic stable envelopes. This construction allows us to relate various representation theoretic objects of   $K(\textsf {X})$  , such as action of quantum groups, quantum dynamical Weyl groups,   $R$  -matrices, etc., to those for   $K(\textsf {X}^{!})$  . In particular, we relate the wall   $R$  -matrices of   $\textsf {X}$   to the   $R$  -matrices of the dual variety   $\textsf {X}^{!}$  . As an example, we apply our results to   $\textsf {X}=\textrm {Hilb}^{n}({{\mathbb {C}}}^2)$  —the Hilbert scheme of   $n$   points in the complex plane. In this case, we arrive at the conjectures of Gorsky and Negut from [ 10 ].

Pursuing Quantum Difference Equations II: 3D mirror symmetry