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

We resolve a conjecture of de Saint Germain: all Y-friezes arise from ${\mathrm{S}\mathrm{L}}_2$-friezes. Additionally, we give a threefold characterisation of those ${\mathrm{S}\mathrm{L}}_2$-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.