We can also take $Hilb$ 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$Fin Hilb$ of finite-dimensional Hilbert spaces becomes a dagger compact category.

Note that either way, the core (of isomorphisms in the first case, or of unitary isomorphisms in the other case) is the same groupoid, whose morphisms are all invertible linear maps of norm exactly $1$.