  Adjoint functors can be thought of as being conceptual inverses. This is probably best illustrated by an example. Let \(U: \mathbf{Grp} \rightarrow \mathbf{Set}\) be the forgetful functor, that is, the functor that sends to each group \(G\) its underlying set of elements \(U(G)\), and to a group homomorphism \(\boldsymbol{f}: G \rightarrow H\) the underlying set function \(U(\boldsymbol{f}): U(G) \rightarrow U(H).\) In other words, \(U\) forgets about the group structure and forgets the fact that morphisms are group homomorphisms. The categories \(\mathbf{Grp}\) and \(\mathbf{Set}\) are certainly not isomorphic, as categories, to one another. (A simple argument runs as follows: the category \(\mathbf{Grp}\) has a zero object, whereas \(\mathbf{Set}\) does not.) Thus, we certainly cannot find an inverse, in the usual algebraic sense, to the functor \(U.\) But there are many non-isomorphic ways to define a group structure on a given set \(X\), and one might hope that among these constructions at least one is functorial and systematically related to the functor \(U.\) What is the conceptual inverse to the operation of forgetting all the group theoretical structure and obtaining a set? It is to construct a group from a set solely on the basis of the concept of group and nothing else, i.e., with no extraneous relation or data. Such a group is constructed “freely”; that is, with no restriction whatsoever except those imposed by the axioms of the theory. In other words, all that is remembered in the process of constructing a group from a given set is the fact that the resulting construction has to be a group. Such a construction exists; it is functorial and it yields what are called free groups. In other words, there is a functor \(F: \mathbf{Set} \rightarrow \mathbf{Grp}\), which to any set \(X\) assigns the free group \(F(X)\) on \(X\), and to each function \(\boldsymbol{f}: X \rightarrow Y\), the group homomorphism \(F(\boldsymbol{f}): F(X) \rightarrow F(Y)\), defined in the obvious manner. The situation can be described thusly: we have two conceptual contexts, a group theoretical context and a set theoretical context, and two functors moving systematically from one context to the other in opposite directions. One of these functors is elementary, namely the forgetful functor \(U.\) It is apparently trivial and uninformative. The other functor is mathematically significant and important. The surprising fact is that \(F\) is related to \(U\) by a simple rule and, in some sense, it arises from \(U.\) One of the striking features of adjoint situations is precisely the fact that fundamental mathematical and logical constructions arise out of given and often elementary functors.