$\newcommand{\domain}[1]{\operatorname{dom} #1}$
Recursive functions have been a topic of importance in computer science. In lambda calculus, the class of recursive functions, $\mathcal{R}$, is defined using initial functions and three kinds of closures. Specifically, we have the following three types of initial function, namely, $U$, $S^{+}$ and $Z$. In simple words, $U^{n}_{i}$ takes the $i$-th element of an $n$-tuple, $S^{+}$ takes the successor of a natural number, and $Z$ returns zero for all natural numbers.
In verbose formal language, $U$ is a map such that for any $i \leq_{\mathbb{N}} n$, $U\left(n, i\right)$ is a function such that for any $x \in \mathbb{N}^{n}$, \begin{equation} U\left(n,i\right)\left(x\right) = x_{i}. \end{equation} For simplicity, we write \begin{equation} U^{n}_{i}\left(x\right) = x_{i}. \end{equation}
Further, $S^{+}: \mathbb{N} \to \mathbb{N}$ is a function such that for any $n \in \mathbb{N}$, \begin{equation} S^{+}\left(n\right) = n + 1. \end{equation} Lastly, for any $n \in \mathbb{N}$, \begin{equation} Z\left(n\right) = 0. \end{equation}
Further, we have the following three types of closures, namely, closure under composition, closure under primitive recursion and closure under minimalization:
$\mathcal{A}$ is closed under composition if and only if $\mathcal{A}$ is a class of numeric functions and for any $n \in \mathbb{N}$, if $x \in \mathcal{A}$ and $\domain{x} = \mathbb{N}^{n}$ and $\psi$ is a sequence such that for any $i \in \mathbb{N}$, $\psi_{i} \in \mathcal{A}$ and $\domain{\psi} = \mathbb{N}^{n}$, if $\varphi$ is a function such that $\domain{\varphi} = \mathbb{N}^{n}$ and for any $\boldsymbol{n} \in \mathbb{N}^{n}$, \begin{equation} \varphi\left(\boldsymbol{n}\right) = x\left(\psi_{1}\left(\boldsymbol{n}\right), \dots, \psi_{n}\left(\boldsymbol{n}\right)\right), \end{equation} then $\varphi \in \mathcal{A}$.
$\mathcal{A}$ is closed under primitive recursion if and only if $\mathcal{A}$ is a class of numeric functions and for any $n \in \mathbb{N}$, $x, \psi \in \mathcal{A}$ such that $\domain{x} = \mathbb{N}^{n}$ and $\domain{\psi} = \mathbb{N}^{n + 2}$, if $\varphi$ is a numeric function such that $\domain{\varphi} = \mathbb{N}^{n + 1}$ and for any $\boldsymbol{n} \in \mathbb{N}^{n}$, \begin{equation} \varphi\left(0, \boldsymbol{n}\right) = x\left(\boldsymbol{n}\right) \end{equation} and \begin{equation} \varphi\left(k + 1, \boldsymbol{n}\right) = \psi\left(\varphi\left(k, \boldsymbol{n}\right), k, \boldsymbol{n}\right), \end{equation} then $\varphi \in \mathcal{A}$.
$\mathcal{A}$ is closed under minimization if and only if $\mathcal{A}$ is a class of numeric functions, and for any $n \in \mathbb{N}$, for any $x \in \mathcal{A}$ such that $\domain{x} = \mathbb{N}^{n + 1}$ and for any $\boldsymbol{n} \in \mathbb{N}^{n}$, there exists some $m \in \mathbb{N}$ such that $x\left(\boldsymbol{n}, m\right) = 0$, for any $\varphi$, if $\domain{\varphi} = \mathbb{N}^{n}$ and for any $\boldsymbol{n} \in \mathbb{N}^{n}$, \begin{equation} \varphi\left(\boldsymbol{n}\right) = \mu m \left[x\left(\boldsymbol{n},m\right) = 0\right], \end{equation} where $\mu m \left[x\left(\boldsymbol{n},m\right) = 0\right]$ is the least element $m$ such that $x\left(\boldsymbol{n}, m\right) = 0$, then $\varphi \in \mathcal{A}$.
Finally, we have the definition for $\mathcal{R}$:
$\mathcal{R}$ is the class of recursive functions if and only if it is the smallest class of numeric functions that contains all initial functions and is closed under composition, primitive recursion and minimalization.
It is my understanding that in theory, we should be able to construct any recursive function of common sense from the three initial functions and the three closure operations, e.g., we should be able to construct the factorial function, namely,
\begin{equation} f\left(n\right) = \begin{cases} 1,\ n = 0\\ n \cdot f\left(n - 1\right),\ n > 0 \end{cases}, \end{equation}
from initial functions and closure operations. To explain, intuitively, $f \in \mathcal{R}$, and $f$ is not any of the three initial functions, so $f$ must can be constructed from closure operations. However, I am not sure how to describe the construction. Can someone help me to understand it?