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:

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

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

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

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

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