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

$\mathop{\mathrm{Mor}}\nolimits (W, U \times V) = \mathop{\mathrm{Mor}}\nolimits (W, U) \times \mathop{\mathrm{Mor}}\nolimits (W, V)$

for any third cosimplicial object $W$ of $\mathcal{C}$.

Proof. Omitted. $\square$

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