This is question regarding Silverman's 'the arithmetic of elliptic curves'(AEC), p325, theorem 3.6.
I want prove there is bijection between $WC(E/K)$ and $H^1(G_K,E)$.
The bijection is given by {$C/K$}→{$σ→p_0^σ-p_0$}.
To prove injectivity, Suppose copyables $p_0^σ-p_0$ and $p_0'^σ-p_0'$ corresponding to $C/K$ and $C'/K$ are cohomologous.
Then, AEC reads $θ:C→C', θ(p)＝p_0'+(p-p_0) +P_0$ gives the isomorphism.
AEC reads 'It is clear that θ is a $\overline{K}$-isomorphism'.
My question is ; Why θ is a $\overline{K}$-isomorphism ? I tried to find inverse, but the middle one $+(p-p_0)$ is not isomorphism, so I'm confused.