Let $$ \mathbf{Grp} $$ be the category of groups and group homomorphisms, and $$ \mathbf{Set} $$ the category of sets. Define a functor $$ U : \mathbf{Grp} \to \mathbf{Set} $$ as follows. Objects $$ U(G) = the underlying set of G. $$ Morphisms For a group homomorphism $$ f : G \to H $$ define $$ U(f) = f $$ viewed as a function between sets. This functor is called the forgetful functor.