if we interpret the top arrow as a diagram in ba and the bottom triangle as a diagram in set, then this diagram properly expresses that every function "f" : "x" → "ub" extends to a unique boolean algebra homomorphism "f"′ : "fx" → "b". the functor "u" can be thought of as a device to pull the homomorphism "f"′ back into set so it can be related to "f".