Strongly ribbon category

A strongly ribbon category is a ribbon category $\mathbb{C}$ equipped with a dagger structure such that the functor $\dagger�: \mathbb{C}^{op} \rightarrow \mathbb{C}$ coherently preserves the ribbon structure.

References

S. Abramsky and B. Coecke, A categorical semantics of quantum protocols, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004).
E.O. Paquette, A categorical semantics for topological quantum computation, Master's thesis, University of Ottawa (2004).