As with many algebraic structures, the conditions and data which define a monoid structure on a set can be reframed diagrammatically, leading to the notion of a monoid object in a monoidal category. Dualizing this definition provides the notion of a comonoid. In this talk, we introduce and discuss monoids and comonoids in SMCs. These structures can be used to prove that any category with a notion of uniform deleting or uniform copying (or cloning) cannot describe quantum theory.
References:
1. Heunen, Chris, and Jamie Vicary. Categories for Quantum Theory: an introduction. Oxford
University Press, 2019. Chapter 4.