Forgetful functors tend to have left adjoints, which are 'free' constructions. For example:
For a more extensive list, see (Mac Lane 1997).
As this is a fundamental example of adjoints, we spell it out:
adjointness means that given a set X and an object (say, an R-module) M, maps of sets correspond to maps of modules : every map of sets yields a map of modules, and every map of modules comes from a map of sets.
In the case of vector spaces, this is summarized as:
"A map between vector spaces is determined by where it sends a basis, and a basis can be mapped to anything."
Symbolically:
The unit of the free–forgetful adjunction is the "inclusion of a basis": .
Fld, the category of fields, furnishes an example of a forgetful functor with no adjoint. There is no field satisfying a free universal property for a given set.
