algebra 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. See also Adjoint functors Functors Projection (set theory) References Mac Lane, Saunders. Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, Berlin, Heidelberg, New York, 1997. Functors