Ring automorphisms between cluster algebras of finite type $A$

Question

To construct a cluster algebra (of finite type $A$) associated to a convex $m$-gon $P_m$ we first take a triangulation $T$ of $P_m$ and regard it as our initial seed with exchangeable variables being the diagonals in $T$ and with the frozen variables being the edges of $P_m$. I read that if we start with a different triangulation $T'$ of $P_m$ we will end up with an isomorphic (as rings?) cluster algebra. Say I want to construct a map which extends to a ring automorphism between cluster algebras associated to $P_m$ w/ initial seed $T$ and cluster algebra associated to $P_m$ w/ initial seed $T'$. Is it enough to show it maps all exchangeable variables to exchangeable variables and all frozen to frozen, bijectively?