  We show that the Fock-Goncharov quantum monodromy matrices satisfy the
relations of the quantum special linear group $\mathrm{SL}_n^q$. The proof
employs a quantum version of the technology invented by Fock and Goncharov
called snakes. This relationship between higher Teichm\"uller theory and
quantum group theory is integral to the construction of a
$\mathrm{SL}_n(\mathbb{C})$-quantum trace map for links in thickened surfaces,
developed in a companion paper (arXiv:2101.06817).
