The Stacks project

Definition 14.9.1. Let $\mathcal{C}$ be a category. Let $U$ and $V$ be cosimplicial objects of $\mathcal{C}$. Assume the products $U_ n \times V_ n$ exist in $\mathcal{C}$. The product of $U$ and $V$ is the cosimplicial object $U \times V$ defined as follows:

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

2. for any $\varphi : [n] \to [m]$ the map $(U \times V)(\varphi ) : U_ n \times V_ n \to U_ m \times V_ m$ is the product $U(\varphi ) \times V(\varphi )$.