sanctuary-type-classes | : rainbow : Standard library for Fantasy Land

I am a student of functional programming, sorry if my question sounds weird--I am trying to wrap my mind around the given type signatures for functions and how they are implemented.

Looking at the signature for ap (Substitution)

https://gist.github.com/Avaq/1f0636ec5c8d6aed2e45

(a → b → c) → (a → b) → a → c

Is given here as

const S = f => g => x => f(x)(g(x));

Which I think I understand. f is a function that takes two parameters, a and b and returns c. g is a function that takes a and returns b. So g(a) returns b and therefore f(a)(b) can be written as f(a)(g(a)), which returns c.

g(a) is the substitute for b ?

Ok now I'm looking at a different implementation that still makes sense:

https://github.com/sanctuary-js/sanctuary-type-classes/tree/v7.1.1#ap--applyf--fa-bfa---fb

ap(Identity(Math.sqrt), Identity(64))

The type signature

1. (f (a -> b), f a) -> f b

Seem similar to

1. (a → b → c) → (a → b) → a → c

Re-writing the second using a = f, b = a, and c = b I get

1. (f -> a -> b) -> (f -> a) -> f -> b

Presuming that ap takes two parameters, where in the first f could be some functor that contains a function a -> b and in the second f some functor that contains a returning a functor that substitutes the first functor's function with the end point b given then functor containing a.

Okay, stepping back, these two things looks vastly different and I can't wrap my mind around how they are somehow saying the same thing.

1. const S = f => g => x => f(x)(g(x))

2. ap(Identity(Math.sqrt), Identity(64))

From my understanding, ap(F(g),F(a)) can be express as F(a).map(g) which, again, I still have a hard time equating to const S = f => g => x => f(x)(g(x)). Perhaps I'm misunderstanding something.

...maybe my misunderstanding has to do with the expression of ap and how that correlates to f => g => x => f(x)(g(x)) because I can see how they both express the same signature but I don't see them as the same thing.

Anyone who can lend some cognitive assistance here, I would greatly appreciate it