Placement of ⊙-bicharades within known mathematical structures

Determine the mathematical classification or conceptual context of a ⊙-bicharade, defined by an object B in a symmetric monoidal category together with a morphism E: B⊗B → B⊗B satisfying (1⊗E)(E⊗1)=(E⊗1)(1⊗σ)(E⊗1)(1⊗σ)(1⊗E), where σ is the symmetry. Clarify whether such structures correspond to any established algebraic or categorical notions or constitute a new class.

Background

Example 29 treats the terminal category ⊙, which is strict bicharadic. In this case, a ⊙-bicharade reduces to a single object B with a self-map E on B⊗B subject to a nontrivial coherence identity involving the symmetry σ.

The authors explicitly remark that they lack a known framework or classification for this object, thereby posing a conceptual question about its nature and connections with familiar structures (e.g., braidings, Yang–Baxter-type solutions, or other categorical symmetries).