one shortcoming of the above definition is that the diagram doesn't capture that "f"′ is a homomorphism; since it is a diagram in set each arrow denotes a mere function. we can fix this by separating it into two diagrams, one in ba and one in set. to relate the two, we introduce a functor "u" : ba → set that "forgets" the algebraic structure, mapping algebras and homomorphisms to their underlying sets and functions.