to if we define for any group homomorphism . Note that is indeed a group homomorphism from to : The content of the above statement is: "The identity functor is naturally isomorphic to the opposite functor " To prove this, we need to provide isomorphisms for every group , such that the above diagram commutes. Set . The formulas and show that is a group homomorphism with inverse . To prove the naturality, we start with a group homomorphism and show , i.e. for all in . This is true since and every group homomorphism has the property . Abelianization Given a group , we can define its abelianization . Let denote the projection map onto the cosets of . This homomorphism is "natural in ", i.e., it defines a natural transformation, which we now check. Let be a group. For any homomorphism , we have that is contained in the kernel of , because any homomorphism into an abelian group kills the commutator subgroup. Then factors through as for the unique homomorphism . This makes a functor and a natural transformation, but not a natural isomorphism, from the identity functor to . Hurewicz homomorphism Functors and natural transformations abound in algebraic topology, with the Hurewicz homomorphisms serving as examples. For any pointed topological space and positive integer there exists a group homomorphism from the -th homotopy group of to the -th homology group of . Both and are functors from the category Top* of pointed topological spaces to the category Grp of groups, and is a natural transformation from to . Determinant Given commutative rings and with a ring homomorphism , the respective groups of invertible matrices and inherit a homomorphism which we denote by , obtained by applying to each matrix entry. Similarly, restricts to a group homomorphism , where denotes the group of units of . In fact, and are functors from the category of commutative rings to . The determinant on the group , denoted by , is a group homomorphism which is natural in : because the determinant is defined by the same formula for every ring, holds. This makes the determinant a natural transformation from to . Double dual of a vector space If is a field, then for every vector space over we have a "natural" injective linear map from the vector space into its double dual. These maps are "natural" in the