So I am struggling with some notation in this book. I tried to draw a diagram to clear up where things are being taken, but the relevant arrow shows up twice, which to me is confusing. How is one supposed to reconcile this? Perhaps I am drawing it wrong. My guess is that we consider the elements $a, b$ as zeroes of $f$ in the first arrow going down, and then as field elements in the arrow going up to the right. Does this make things better?
The relevant image:
https://ibb.co/jTZbprP