The idea is that f(a,b) = c iff there is a sequence of pairs of numbers <<f(0,b),0>, <f(1,b),1>, ..., <f(a,b),a>> where the first element is <g(b),0>, the last element is <c,a>, and each subsequent element is obtained from the previous one using h. You only need to prove that the set of such sequences is representable; then you can use the minimization operator to express f (slide 20).

It is possible to write a formula that takes n and says that n is the code of a sequence as above. The important thing is that all quantifiers can be bounded by n because all numbers involved (e.g., each sequence component, the length of the sequence, etc.) are <= than the code of the sequence.

The idea of having a sequence of pairs <f(i,b),i> and not just a sequence of numbers f(i) comes from the fact that h takes the counter as an argument in addition the previous value of f.