Proof: Applications of the Universal Property of Natural Numbers

1. The problem statement, all variables and given/known data
N refers to the set of all natural numbers.
Part 1: There are two different parts I'm having trouble with:
Let S be a set and let f : S → S be a function. Show that
for any n ε N, there exists a function denoted by f^n : S → S
such that f^1 = f and f^σ(n) = f ° fn.

Part 2: From the previous problem, we have σ^n : N → N for all n ε N.
Show that for any n ε N, σ^(n+1) of (N) is a subset of σ^n of (N), where we have
used n + 1 for σ(n) as we defined in class.

2. The attempt at a solution
Part 1: For my reasoning, we have to prove the two given statements, f^1 = f and f^σ(n) = f ° f^n. So f^1 is clearly equivalent to f. After, the only thing I could think of doing is letting σ(n) = n+1, using the property of commutativity of addition, to have f^(n+1)= f ° f^n => f^(1+n) = f° f^n => f ° f^n = f ° f^n, though I feel as if this is not valid reasoning. Beforehand, I proved this for an element, say x ε N, but can construct functions to navigate around the elements. How do I do this for functions?

For Part 2, I believe the goal would be to prove that given any x ε σ^(n+1) of (N), that x ε σ^n of (N) as well. For this problem, I am not sure where to start for this problem, since it seems like it would be the opposite direction (the subset would be the other way).

I'm having trouble understanding this notation.

Does f^σ(n) mean f^σ(n) or f^σ(n)?

Also, what is fn? Do you mean fⁿ?

I have a suspicion that this is about proofs by mathematical induction.