We introduce a (categorical) Fraïssé theory applied to the category of abstract Cuntz semigroups, drawing inspiration from Kubiś' approach for metric-enriched categories. Throughout the discussion, we will explore metric properties available within the Hom-sets of Cu-morphisms, including concepts like Cauchy sequences and approximate intertwinings. Lastly, we will illustrate instances of Cu-semigroups that naturally emerge as Fraïssé limits.

This is joint work with E. Vilalta.