I saw this post recently: Tannaka–Krein duality I have this question please: in the following which i report here:

"The problem is with surjectivity: let us denote G:=G(R(G)) and ^:R(G)→R(G) is an isomorphism from the previous theorem. One shows that δ∗:F↦F∘δ is right inverse of ^ and therefore is also an isomorphism. Since R(G) and R(G) are dense in C(G) and C(G) resp. it follows that δ∗ extends to the isomomorhism between C(G) and C(G) and thus δ is surjective."

Why if δ∗ is right inverse of ^ then it is also an isomorphism? Have left invertibilty can be proved? And can we use only algebraic arguments on G?