Of course we should define the product of cosimplicial objects as the product in the category of cosimplicial objects. This may lead to the potentially confusing situation where the product exists but is not described as below. To avoid this we define the product directly as follows.
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:
$(U \times V)_ n = U_ n \times V_ n$,
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 $U(\varphi ) \times V(\varphi )$.
Lemma 14.9.2. If $U$ and $V$ are cosimplicial objects in the category $\mathcal{C}$, and if $U \times V$ exists, then we have for any third cosimplicial object $W$ of $\mathcal{C}$.
\[ \mathop{\mathrm{Mor}}\nolimits (W, U \times V) = \mathop{\mathrm{Mor}}\nolimits (W, U) \times \mathop{\mathrm{Mor}}\nolimits (W, V) \]
Proof. Omitted. $\square$
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