Interpretation and classification of Arr(M)-bicharades

Classify and conceptually interpret Arr(M)-bicharades for an associative monoid M, defined as collections B={B(a)}_{a∈M} of objects with structure operations B(a)⊗B(b) → B(c)⊗B(d) specified whenever ab=cd. Identify whether these structures correspond to known algebraic or categorical frameworks or introduce a new class.

Background

Example 30 considers the arrow category Arr(M) of a monoid M as a strict bicharadic category. An Arr(M)-bicharade comprises a family of objects indexed by M with operations constrained by the monoid equation ab=cd.

Despite specifying the operations, the authors note they cannot situate these objects within a known theory, indicating an unresolved question regarding their interpretation and classification.