Topologists know how to cook up a group called the fundamental group from any space. But ALSO, any map between spaces determines a homomorphism of the fundamental groups. So the fundamental group is really a functor from the category Top to the category Group. For example, it's easy to use this idea to show that you can't map a disk continuously to its boundary in such a way that the points on the boundary get mapped to themselves. No one is a true mathematical physicist unless they can explain that remark.
First quantization is a mystery.
Now it doesn't seem to be true that God created a classical universe on the first day and then quantized it on the second day. So it's unnatural to try to get from classical to quantum mechanics. Nonetheless we are inclined to do so since we understand classical mechanics better. It has become clear that there is no utterly general systematic procedure for doing so. Alas, there is no such nice functor.