Naturality of SL₃ quantum trace maps for surfaces

Fock-Goncharov's moduli spaces X_ PGL₃,S of framed PGL₃-local systems on punctured surfaces S provide prominent examples of cluster X-varieties and higher Teichm\"uller spaces. In a previous paper of the author (arXiv:2011.14765), building on the works of others, the so-called SL₃ quantum trace map is constructed for each triangulable punctured surface S and an ideal triangulation Δ of S, as a homomorphism from the stated SL₃-skein algebra of the surface to a quantum torus algebra that deforms the ring of Laurent polynomials in the cube-roots of the cluster coordinate variables for the cluster X-chart for X_ PGL₃,S associated to Δ. We develop quantum mutation maps between special subalgebras of the cube-root quantum torus algebras for different triangulations and show that the SL₃ quantum trace maps are natural, in the sense that they are compatible under these quantum mutation maps. As an application, the quantum SL₃-PGL₃ duality map constructed in the previous paper is shown to be independent of the choice of an ideal triangulation.