Yes! You are right. In adfition the two arrows are in the wrong direction.
You can also prove the two directions at once eval not injective iff there is a non trivial element p in the kernel, if there is a non zero poly p that vanishes if evaluated at alpha.. so this not really a bug deal.
This is exactly the reason I do not usually make my notes public: for a proof like that that I could do I woken up in the middle of the night, I write them down just to estimate the time I will need to present them (I know 2x45 minutes leq 4 pages of notes) without taking care of them being perfect/correct (as you are starting of discovering with my swapping of Fs and KS I do not always write what my brain means) :)