The Stanford Encyclopedia of Philosophy's article on category theory claims that adjoint functors can be thought of as "conceptual inverses" of each other.
For example, the forgetful functor "ought to be" the "conceptual inverse" of the free-group-making functor. Similarly, in multigrid the restriction operator "ought to be" the conceptual inverse of it's adjoint prolongation operator.
I think there is some deep and important intuition here, but so far I can only grasp it in specific cases and not in the abstract sense. Can anyone help shed light on what is meant by this statement about adjoint functors being conceptual inverses?