Explanation of a passage in Atiyah / Macdonald

Question

On page 105 the authors show that $\hat{\hat{G}} \cong \hat{G}$ (Proposition 10.5) and conclude that the canonical homomorphism $\phi : \hat{G} \to \hat{\hat{G}}$ is an isomorphism. How does the fact that $\hat{G}$ and $\hat{\hat{G}}$ are isomorphic imply that the particular map $\phi$ is an isomorphism?