Define a length function over $A^{*} \leftarrow{N}$ such that $length(l)$ outputs the length of $l$
Let F denote a “shape”, as your LIST and NAT shapes for example.


  Declaring $I := μF$ is tantamount to saying that $I$ is recursively defined by some $α : F(I) ≅ I$
    and moreover for any $β : F(K) → K$ there is a unique $f : I → K$ that respects the “F”-structure.


Since you define $A^* = μ(LIST)$, where the “LIST” structure on a set $X$ consists of a constant $c : X$
and a function $n : A × X → X$ --- c.f., your $nil, -·-$ --- we automatically obtain an operation $A^* → ℕ$ if
we can endow $ℕ$ with a “LIST” structure; namely a constant such as zero $0 : ℕ$ and an operation
on ℕ such as $A \times ℕ → ℕ : (a, n) ↦ s\, n$.

So by the μ-definition of lists, since we furnished ℕ with a LIST structure,
we have that there is a unique function, call it $length$, that respects the LIST structure in that
it takes the shape constants-to-constants and the shape functions-to-functions:
$$ \mathsf{length}(nil) \;=\; 0 $$
$$ \mathsf{length}(a \cdot l) \;=\; s(\mathsf{length}\, l) $$

Alternatively, these properties also serve to “define length by structural induction on  the shape of lists”.

Hope that helps a bit :-)