	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".