 We now consider how free algebras are organized from the perspective of category theory. We defined the free algebra \(B\) generated by a subset \(X\) of \(B\) as having the property that for every algebra \(A\) and every valuation \(f: X\rightarrow A\), there exists a unique homomorphism \(h: B\rightarrow A\). Now every homomorphism \(h: B\rightarrow A\) necessarily arises in this way, since its restriction to \(X\), as a function from \(X\) to \(A\), is a valuation. Furthermore every function \(f: X\rightarrow A\) arises as the restriction to \(X\) of its extension to a homomorphism. Hence we have a bijection between the functions from \(X\) to \(A\) and the homomorphisms from \(B\) to \(A\).