Let be a concrete category and T: S the forgetful functor. If T has

Let ℓ be a concrete category and T: ℓ → S the forgetful functor. If T has a left adjoint F : S → ℓ, then F is called a free-object functor and F(X) (X ϵ S) is called a free F-object on X.

(a) The category of groups has a free-object functor.

(b) The category of commutative rings with identity and identity preserving homomorphisms has a free-object functor.