On page 204 of Rick Miranda's Algebraic Curves and Riemann Surfaces, he talks about how the canonical map of a hyperelliptic curve is the double cover map composed with the Veronese map $\phi(x) = [1:x:\cdots:x^{g-1}]$. Then, he says:
the double covering map for a hyperelliptic curve of genus $g\ge 2$ is unique since is it the canonical map after all.
I don't understand what is meant by unique and how this follows from the discussion.