 By way of proof, pick any bijection \(f: X\rightarrow Y\). This, its inverse \(f': Y\rightarrow X\), and the two identity functions on respectively \(X\) and \(Y\), form a system of four functions closed under composition. Each of these functions is from a generator set to an algebra and therefore has a unique extension to a homomorphism. These four homomorphisms are also closed under composition. The one from \(B\) to itself extends the identity function on \(X\) and therefore must be the identity homomorphism on \(B\) (since the latter exists and its restriction to \(X\) is the identity function on \(X)\). Likewise the homomorphism from \(G\) to \(G\) is an identity function. Hence the homomorphisms between \(B\) and \(G\) compose in either order to identities, which makes them isomorphisms. But this is what it means for \(B\) and \(B'\) to be isomorphic.