[Edited]
Let $\mathsf{PR}$ be the category defined as follows:
Choose a specific presentation of Primitive Recursive Arithmetic, that is, with a specific set of terms for primitive recursive functions.
Let the objects of $\mathsf{PR}$ be any $1$-ary term $A(x)$ in PRA. (Intuitively, think of a primitive recursive set given explicitly as as as set of zeros of some term).
Let the morphisms $\phi: A \rightarrow B$ of $\mathsf{PR}$ be a $1$-ary terms $\varphi(x)$ such that $\vdash_{PRA} A(x) = 0 \Rightarrow B(\varphi((x)) = 0$. Identify two morphisms $\varphi$ and $\psi$ if PRA proves $\varphi(x) = \psi(x)$.
Composition of morphisms is composition of terms.
Question: does this category have other names or other presentations? It seems like this category should be of interest.
Any references would be appreciated.