  The fact that \(U\) and \(F\) are conceptual inverses expresses itself formally as follows: applying \(F\) first and then \(U\) does not yield the original set \(X\), but there is a fundamental relationship between \(X\) and \(UF(X).\) Indeed, there is a function \(\eta\): \(X \rightarrow UF(X)\), called the unit of the adjunction, that simply sends each element of \(X\) to itself in \(UF(X)\) and this function satisfies the following universal property: given any function \(\boldsymbol{g}: X \rightarrow U(G)\), there is a unique group homomorphism \(\boldsymbol{h}: F(X) \rightarrow G\) such that \(U(\boldsymbol{h})  \circ \eta = \boldsymbol{g}.\) In other words, \(UF(X)\) is the best possible solution to the problem of inserting elements of \(X\) into a group (what is called “insertion of generators” in the mathematical jargon). Composing \(U\) and \(F\) in the opposite order, we get a morphism \(\xi : FU(G) \rightarrow G\), called the counit of the adjunction, satisfying the following universal property: for any group homomorphism \(\boldsymbol{g}: F(X) \rightarrow G\), there is a unique function \(\boldsymbol{h}: X \rightarrow U(G)\) such that \(\xi \circ F(\boldsymbol{h}) = \boldsymbol{g} \circ FU(G)\) constitutes the best possible solution to the problem of finding a representation of \(G\) as a quotient of a free group. If \(U\) and \(F\) were simple algebraic inverses to one another, we would have the following identity: \(UF = I_{\mathbf{Set}}\) and \(FU = I_{\mathbf{Grp}}\), where \(I_{\mathbf{Set}}\) denotes the identity functor on \(\mathbf{Set}\) and \(I_{\mathbf{Grp}}\) the identity functor on \(\mathbf{Grp}.\) As we have indicated, these identities certainly do not hold in this case. However, some identities do hold: they are best expressed with the help of the commutative diagrams: