The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let ϕ ϕ : : G → → H be a Lie group homomorphism and let ϕ ϕ ∗ ∗ be its derivative at the identity. If we identify the Lie algebras of G and H with their tangent spaces at the identity elements, then ϕ ϕ ∗ ∗ is a map between the corresponding Lie algebras: