In $\text{Thm.} \; 12.4$ in Fomin's and Zelevinsky's paper on coefficients we are given a recipe for constructing a cluster algebra with universal coefficients. The recipe is given in terms of (almost positive) coroots and I am trying to understand how this translates into the usual cluster algebraic type A setup involving convex polygons,triangulations, etc. If I understand correctly, as the initial seed we take any zig-zag triangulation. The coefficient field is the Tropical semifield (freely) generated by dual diagonals, I think. For instance, in $A_2$, we'd have that $\mathbb{P}=\text{Trop}(x_{12}x_{34},x_{12}x_{45},x_{23}x_{45},x_{23}x_{15},x_{34}x_{15})$.In other words, each generator represents a pair of "disjoint" sides of a pentagon. Now, I'm not sure about the next part. It is probably something easy but I am failing to see the geometric description of the $Y$-coefficients in the initial coefficient tuple ($12.5$ part of $\text{Thm.} \; 12.5$). Any help would be very much appreciated.

In Thm. 12.4 in Fomin and Zelevinsky - Cluster algebras IV: Coefficients we are given a recipe for constructing a cluster algebra with universal coefficients. The recipe is given in terms of (almost positive) coroots and I am trying to understand how this translates into the usual cluster algebraic type $A$ setup involving convex polygons, triangulations, etc. If I understand correctly, as the initial seed we take any zig-zag triangulation. The coefficient field is the tropical semifield (freely) generated by dual diagonals, I think. For instance, in $A_2$, we'd have that $\mathbb{P}=\operatorname{Trop}(x_{12}x_{34},x_{12}x_{45},x_{23}x_{45},x_{23}x_{15},x_{34}x_{15})$. In other words, each generator represents a pair of "disjoint" sides of a pentagon. Now, I'm not sure about the next part. It is probably something easy but I am failing to see the geometric description of the $Y$-coefficients in the initial coefficient tuple (12.5 part of Thm. 12.5). Any help would be very much appreciated.