We can also take to be the dagger category with the same object but whose morphisms are now all bounded linear maps between them. (The dagger operation is the usual adjoint operation on such maps.) The full subcategory of finite-dimensional Hilbert spaces becomes a dagger compact category.
In any case, the forgetful functor from to Vect is faithful, confirming the intuition that a Hilbert space is a vector space equipped with extra structure. is also a full subcategory of Ban, the category of Banach spaces.
A pedagogical description of the monoidal structure on with an emphasis on their role in quantum mechanics and their relation to nCob is in