In a topos with a natural numbers object $(𝑁,𝑧,𝑠)$, you can define finite cardinals as objects that arise as the pullback along a morphism $𝑝:\top \to 𝑁$ of the generic finite cardinal. Write $[𝑝]$ for the cardinal corresponding to $𝑝$.

The above characterization is a little obtuse and does warrant some more explanation. One way to make it more concrete is that $[𝑧]=\perp$ and $[𝑠\circ 𝑛]=\top \oplus [𝑛]$, and that finite cardinals warrant a nice induction principle. If $𝑃$ is a property expressible in the internal language such that $\perp$ satisfies $𝑃$, and that whenever $𝐴$ satisfies $𝑃$ then $\top \oplus 𝐴$ satisfies $𝑃$; then every finite cardinal satisfies $𝑃$. That is, $𝑃$ forms a $(𝑧,𝑠)$-closed subobject of $𝑁$ and thus $𝑃$ is all of $𝑁$.

My working mental model is in a presheaf topos, where the natural numbers object can be defined explicitly as $\mathrm{\Delta }(\mathbb{N})$.

Definition 1. Constant Presheaf constant-presheaf

The constant presheaf of a set $𝑋$ on a category $𝐶$ is a functor $\mathrm{\Delta }(𝑋):{𝐶}^{𝑜𝑝}\to \text{𝐒𝐞𝐭}$ such that

$$\mathrm{\Delta }(𝑋)𝑐≔𝑋$$

I often call this the discrete presheaf for $𝑋$, but I don’t know if that’s standard.

The finite cardinals with respect to $\mathrm{\Delta }(\mathbb{N})$ can be characterized then as,

These obey nice algebraic properties.