	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.