Dagger symmetric monoidal category

In the mathematical field of category theory, a dagger symmetric monoidal category is a monoidal category formula_1 that also possesses a dagger structure. That is, this category comes equipped not only with a tensor product in the category theoretic sense but also with a dagger structure, which is used to describe unitary morphisms and self-adjoint morphisms in formula_2: abstract analogues of those found in FdHilb, the category of finite-dimensional Hilbert spaces. This type of category was introduced by Peter Selinger as an intermediate structure between dagger categories and the dagger compact categories that are used in categorical quantum mechanics, an area that now also considers dagger symmetric monoidal categories when dealing with infinite-dimensional quantum mechanical concepts.
Formal definition.
A dagger symmetric monoidal category is a symmetric monoidal category formula_2 that also has a dagger structure such that for all formula_4, formula_5 and all formula_6 and formula_7 in formula_8,
Here, formula_14 and formula_15 are the natural isomorphisms that form the symmetric monoidal structure.
Examples.
The following categories are examples of dagger symmetric monoidal categories:
A dagger symmetric monoidal category that is also compact closed is a dagger compact category; both of the above examples are in fact dagger compact.