Dagger category

In category theory, a branch of mathematics, a dagger category (also called involutive category or category with involution) is a category equipped with a certain structure called "dagger" or "involution". The name dagger category was coined by Peter Selinger.
Formal definition.
A dagger category is a category formula_1 equipped with an involutive functor formula_2 that is the identity on objects, where formula_3 is the opposite category.
In detail, this means that it associates to every morphism formula_4 in formula_1 its adjoint formula_6 such that for all formula_4 and formula_8,
Note that in the previous definition, the term "adjoint" is used in a way analogous to (and inspired by) the linear-algebraic sense, not in the category-theoretic sense.
Some sources define a category with involution to be a dagger category with the additional property that its set of morphisms is partially ordered and that the order of morphisms is compatible with the composition of morphisms, that is formula_12 implies formula_13, formula_14, formula_15 whenever their sources and targets are compatible.
Remarkable morphisms.
In a dagger category formula_1, a morphism formula_22 is called
The latter is only possible for an endomorphism formula_25. The terms "unitary" and "self-adjoint" in the previous definition are taken from the category of Hilbert spaces, where the morphisms satisfying those properties are then unitary and self-adjoint in the usual sense.