All Y-friezes come from $\mathrm{SL}_2$-friezes

We resolve a conjecture of de Saint Germain: all Y-friezes arise from $\mathrm{SL}_2$-friezes.

Additionally, we give a threefold characterisation of those $\mathrm{SL}_2$-friezes that give rise to the same Y-frieze: an elementary characterisation using quiddities, a combinatorial characterisation using triangulated polygons, and a geometrical characterisation using chains of horocycles.