I know that we can define the constant function as a function $f: \mathbb{N} \rightarrow \mathbb{N}$ by lettting:
$f(0) = c$
$f(n+1) = c$
Let $g = z$ so that $g(n) = 0$ for all $n \in \mathbb{N}$
Let $h = s$
Let $h_1 = \pi_n^n$
Then we can write a recursive definition of the constant function $f: \mathbb{N} \rightarrow \mathbb{N}$ as follows:
$f(0) = h_1(g, s(g), s(s(g)), s(s(s(g)))...)$
$f(n) = h_1(g, s(g), s(s(g)), s(s(s(g)))...)$
Am I on the right track here?