  In this article we show that for every collection $\mathcal{C}$ of an even
number of polynomials, all of the same degree $d>2$ and in general position,
there exist two hyperbolic $3$-orbifolds $M_1$ and $M_2$ with a M\"obius
morphism $\alpha:M_1\rightarrow M_2$ such that the restriction of $\alpha$ to
the boundaries $\partial M_1$ and $\partial M_2$ forms a collection of maps $Q$
in the same conformal Hurwitz class of the initial collection $\mathcal{C}$.
Also, we discuss the relationship between conformal Hurwitz classes of rational
maps and classes of continuous isomorphisms of sandwich products on the set of
rational maps.
