It has become clear that there is no utterly general systematic procedure for doing so. Alas, there is no such nice functor. So quantization is always an ad hoc and problematic thing to attempt. A lot is known about it, but more isn't. That's why first quantization is a mystery.
Every symplectic map yields a unitary operator in an obvious way.
Any unitary operator on H gives a unitary operator on K in an obvious way. This functor is positivity-preserving. By the way, you might enjoy thinking about what happens if you iterate second quantization. Now, an interesting object in physics is Minkowski space. We can also express the principal of general covariance and the principal of gauge-invariance most precisely by saying that observables are functorial.