Forgetful functors between categories of algebraic structures usually have left adjoints. For instance: At several points in this book, we make contact with the idea of an algebraic theory. You already know several examples: the theory of groups is an algebraic theory, as are the theory of rings, the theory of vector spaces over $ \reals $ , the theory of vector spaces over $ \complexes $ , the theory of monoids, and (rather trivially) the theory of sets.