  We introduce a new, algebraic method to construct duality functions for
integrable dynamic models. This method will be implemented on dynamic
stochastic higher spin vertex models, where we prove the duality functions are
the $ _3 \varphi_2$ functions. A degeneration of these duality functions are
orthogonal polynomial dualities of Groenevelt--Wagener arXiv:2306.12318. The
method involves using the universal twister of
$\mathcal{U}_q(\mathfrak{sl}_2)$, viewed as a quasi--triangular,
quasi--$^*$--Hopf algebra. The algebraic method is presented very generally and
is expected to produce duality functions for other dynamic integrable models.
As an application of the duality, we prove that the asymptotic fluctuations of
the dynamic stochastic six vertex model with step initial conditions are
governed by the Tracy--Widom distribution.
