  In this paper, we provide a categorical analysis of the arithmetic theory
$I\Sigma_1$. We will provide a categorical proof of the classical result that
the provably total recursive functions in $I\Sigma_1$ are exactly the primitive
recursive functions. Our strategy is to first construct a coherent theory of
arithmetic $\mathbb T$, and prove that $\mathbb T$ presents the initial
coherent category equipped with a parametrised natural number object. This
allows us to derive the provably total functions in $\mathbb T$ are exactly the
primitive recursive ones, and establish some other constructive properties
about $\mathbb T$. We also show that $\mathbb T$ is exactly the
$\Pi_2$-fragment of $I\Sigma_1$, and conclude they have the same class of
provably total recursive functions.
