According to unpublished notes by Gavin Wraith ("Notes on arithmetic universes and Gödel incompleteness theorems" (1985)), PRA can be described as an equational theory or as a Lawvere theory, and is abstractly characterized as initial among all Lawvere theories whose generating object is a parametrized natural numbers object. (For some details on Wraith's notes, see Alan Morrison's Master's Thesis; see particularly chapter 5.)

This statement may take a bit of unpacking. Lawvere theories are a way of doing universal algebra categorically. Formally, a *Lawvere theory* is a category $T$ with finite products together with a product-preserving functor $\Phi: \text{Fin}^{op} \to T$ that is a bijection on objects. Here $\text{Fin}$ is the category whose objects are finite cardinals $k = \{0, \ldots, k-1\}$ and whose morphisms are functions between them. The cardinal $1$ generates (the objects of) $\text{Fin}$ by taking finite coproducts of copies of $1$; similarly it generates $\text{Fin}^{op}$ under finite products. Thus the special object $x = \Phi(1)$ generates the objects of $T$ by taking finite products: every object of $T$ is of the form $x^n$. General morphisms $x^n \to x^m$ are $m$-tuples of morphisms $x^n \to x$; we think of the morphisms $x^n \to x$ in $T$ as parametrizing the definable $n$-ary operations of an equational theory. [This categorical description of equational theories is closely related to the concept of clone. If you accept that every equational theory gives rise to a finitary monad on $Set$, then the corresponding Lawvere theory is the category opposite to the finitary Kleisli category consisting of finitely generated free objects. But this is a somewhat hurried discussion which I'll cut short here.] A morphism $F: S \to T$ is a product-preserving functor which is compatible with the given product-preserving functors $\text{Fin}^{op} \to S$ and $\text{Fin}^{op} \to T$.

A parametrized natural numbers object in a category with finite products $\mathbf{C}$ is an object $N$ that comes equipped with maps $z: 1 \to N$ (here $1$ denotes the terminal object, and read '$z$' as 'zero') and $s: N \to N$ (successor), such that given any objects $A, X$ of $\mathbf{C}$ and maps $f: A \to X$, $g: X \to X$, there exists a unique map $h: N \times A \to X$ such that the following diagram commutes:

$$\begin{array}{ccc}
A & \stackrel{\langle z \circ !, 1_A\rangle}{\to} & N \times A & \stackrel{s \times 1_A}{\leftarrow} & N \times A \\
& f \searrow & \downarrow h & & \downarrow h \\
& & X & \underset{g}{\leftarrow} & X
\end{array}$$

(here $!$ denotes the unique map $A \to 1$). This axiom is what you need to internalize primitive recursion in a category with finite products.

So now the Lawvere theories we are interested in are those for which the generator $x = \Phi(1)$ is a parametrized natural numbers object. A concrete example of such is the full subcategory of $Set$ whose objects are finite powers $\mathbb{N}^n$ of the set of natural numbers. For that matter, for any category with finite products and a natural numbers object $N$ (for example, a Grothendieck topos), you can cook up a Lawvere theory by considering the full subcategory consisting of finite powers of $N$. Or, the subcategory needn't be full: just retain enough arrows to retain finite product structure and primitive recursive structure guaranteed by the axiom of parametrized NNO's.

Finally, we are interested in the *initial* such Lawvere theory. As is the case with any initial algebraic object, the explicit construction is syntactic: we start with $N$ and $z: 1 \to N$ and $s: N \to N$ and use the axiom of parametrized NNO's together with finite cartesian product structure (products of copies of $N$, projection maps, diagonal maps) and categorical composition to generate formally all the arrows. (As a simple exercise, show how to construct formal addition and formal multiplication on $N$.) One thing to check is that morphisms $1 \to N$ (the definable constants of the equational theory, considered up to provable equality) correspond bijectively to standard natural numbers.

If $T$ is the initial such theory (which according to Wraith is PRA), with generator denoted $N$, and if $f: T \to Set$ is the unique Lawvere theory morphism sending $N$ to $\mathbb{N}$, then the functorial map $\hom_T(N^n, N) \to \hom(f(N^n), f(N)) \cong \hom(\mathbb{N}^n, \mathbb{N})$ is a surjection onto the total $n$-ary primitive recursive functions on the standard natural numbers. Is is an injection? No. For example, consider the primitive recursive "function" $G: N \to N$ defined by $G(n) = 1$ if $n$ codes the proof of a contradiction in ZFC, and $G(n) = 0$ otherwise. (This is primitive recursive because no unbounded searches are required to verify the validity of a proof.) In the standard model of arithmetic living in the ZFC model $Set$, $G$ would be sent to the constant $0$ function mapping $\mathbb{N} \to \mathbb{N}$. But we could equally consider a model $\mathcal{M}$ of $ZFC + \neg Con(ZFC)$ and its natural number object $\mathbb{N}_\mathcal{M}$, in which $G$ would *not* be sent to the constant $0$ function. Thus $G$ and the constant $0$ function must be distinct in the category $T$. (Thanks to Zhen Lin Low for supplying this argument, here.)