 Now the typing here is a little casual, so let us clean it up. Since \(X\) is a set while \(A\) is an algebra, \(f\) is better typed as \(f: X\rightarrow U(A)\) where \(U(A)\) denotes the underlying set of \(A\). And the relationship of \(X\) to \(B\) is better understood with the notation \(B = F(X)\) denoting the free algebra generated by the set \(X\). So \(U\) maps algebras to sets while \(F\) maps sets to algebras. \(F\) and \(U\) are not in general inverses of each other, but they are nonetheless related in a way we now make precise.