We express the fundamental properties of commutative polygroups (also known as canonical hypergroups) in category-theoretic terms, over the category $\mathbf{S}\mathbf{e}\mathbf{t}$ formed by sets and functions. For this, we employ regularity as well as the monoidal structure induced on the category $\mathbf{R}\mathbf{e}\mathbf{l}$ of sets and relations by cartesian products. We highlight how our approach can be generalised to any regular category. In addition, we consider the theory of partial multirings and find fully faithful functors between certain slice or coslice categories of the category of partial multirings and other categories formed by well-known mathematical structures and their morphisms.