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 $\mathcal{G}:=\mathcal{G}(\mathcal{R}(G))$ and $\hat{}:\mathcal{R}(G) \to \mathcal{R}(\mathcal{G})$ is an isomorphism from the previous theorem. One shows that $\delta^*:F \mapsto F \circ \delta$ is right inverse of $\hat{}$ and therefore is also an isomorphism. Since $\mathcal{R}(G)$ and $\mathcal{R}(\mathcal{G})$ are dense in $C(G)$ and $C(\mathcal{G})$ resp. it follows that $\delta^*$ extends to the isomomorhism between $C(G)$ and $C(\mathcal{G})$ and thus $\delta$ is surjective.

Let me please ask the couple of questions below:

1. Why is $\hat{}$ an isomorphism of Hopf algebras?
2. Why if $\delta^*$ is right inverse of $\hat{}$ then it is also an isomorphism? Have left invertibilty can be proved? And can we use only algebraic arguments on $G$?