The Stacks project

Definition 14.6.1. Let $\mathcal{C}$ be a category. Let $U$ and $V$ be simplicial objects of $\mathcal{C}$. Assume the products $U_ n \times V_ n$ exist in $\mathcal{C}$. The product of $U$ and $V$ is the simplicial object $U \times V$ defined as follows:

1. $(U \times V)_ n = U_ n \times V_ n$,

2. $d^ n_ i = (d^ n_ i, d^ n_ i)$, and

3. $s^ n_ i = (s^ n_ i, s^ n_ i)$.

In other words, $U \times V$ is the product of the presheaves $U$ and $V$ on $\Delta $.