Category of Multiples

Let $\mathcal{C}$ be a category where

• the objects are the elements of $\mathbb{N};$
• there is an arrow $(\times m)$ to $mn$ from every positive integer $n \in \mathbb{N}$ , for every positive integer $m \in \mathbb{N}.$

What is the product of $12$ and $20$ in $\mathcal{C}?$

$$
Note: A product of $a, b \in \mathcal{C}$ is an object $a \times b \in \mathcal{C}$ , along with two arrows $p_1 : a \times b \to a$ and $p_2 : a \times b \to b$ , such that for all objects $c \in \mathcal{C}$ the diagram to the right commutes.

---

Inspired by a Categories and Haskell