I've been staring at this for a while, and I'm failing to see why $\phi^*=\iota$ in the proof of (b).
Let $f\in K(C_2)$, I tried to calculate both sides, and got stuck. $$ \phi^*f = f\circ \phi= f([1:\iota(g_1):\dots:\iota(g_N)]) \overset{?}{=} \iota(f) $$ Any help is appreciated! I feel like I'm just misunderstanding a definition or something.
