> Function composition is always associative.

You're still confused. There is no function composition here.

op in the example above is just some other function, like +. The associativity of + and function composition are true for totally unrelated reasons. Associativity of plus is an inductive argument that follows from the Peano axioms.

Function composition says:

(f o g) o h = f o (g o h)

as functions. It is true because unary function application "serializes" function applications. Formally, I mean:

    ((f o g) o h)(x)
      = f(g(h(x)))
      = (f o (g o h))(x)

Function composition has one value flowing through several functions. Fold has several values flowing through one function.