  We show that the quantized Fock-Goncharov 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-Goncharov called
snakes. This relationship between higher Teichm\"uller theory and quantum group
theory is integral to the construction of a $\mathrm{SL}_n$-quantum trace map
for knots in thickened surfaces, developed in a companion paper
(arXiv:2101.06817).
