Involutions of genus 2 curve commute

Question

Let $C$ be a smooth curve of genus 2, and denote by $j:C\to\mathbb{P}^1$ the canonical degree 2 morphism induced by $|K|$, where $K$ is the canonical divisor. Also denote by $\psi:C\to C$ the corresponding covering transformation, which is an involution.

Assume $C$ is bielliptic, i.e. admits a degree 2 morphism $i:C\to E$ to an elliptic curve. Denote by $\varphi:C\to C$ the corresponding covering transformation, which is also an involution. How can I prove that $\varphi$ and $\psi$ commute?

Answer 1

Note that $\psi(P) = K_C - P$ and that $\varphi(K_C) = K_C$. Therefore, $$ \psi(\varphi(P)) = K_C - \varphi(P) = \varphi(K_C - P) = \varphi(\psi(P)). $$