> `a -> b -> f a -> f b`
(a -> b) -> F a -> F b; the parentheses are critical. I don't fully disagree with your point, but if you're going to emphasize rigor and definitions, it would be wise to make sure you're not setting people on the fence up for mistakes akin to those in the original post.
(You also called it `F` and then used it as `f`.)