The dual concept is that of a reflective subcategory. See there for more details.
Vopěnka's principle is equivalent to the statement:
This is (AdamekRosicky, theorem 6.28).
the inclusion of groups into monoids, where the right adjoint takes a monoid to its group of units.
Lie integration, which constructs a simply connected Lie group from a finite-dimensional real Lie algebra. The coreflector is Lie differentiation (taking a Lie group to its associated Lie algebra), and the counit is the natural map to a given Lie group from the universal covering space of the connected component at the identity of .