Lemma 14.19.12. Let $U$, $V$ be $n$-truncated simplicial objects of a category $\mathcal{C}$. Then
\[ \text{cosk}_ n (U \times V) = \text{cosk}_ nU \times \text{cosk}_ nV \]
whenever the left and right hand sides exist.
Proof. Let $W$ be a simplicial object. We have
\begin{eqnarray*} \mathop{\mathrm{Mor}}\nolimits (W, \text{cosk}_ n (U \times V)) & = & \mathop{\mathrm{Mor}}\nolimits (\text{sk}_ n W, U \times V) \\ & = & \mathop{\mathrm{Mor}}\nolimits (\text{sk}_ n W, U) \times \mathop{\mathrm{Mor}}\nolimits (\text{sk}_ nW, V) \\ & = & \mathop{\mathrm{Mor}}\nolimits (W, \text{cosk}_ n U) \times \mathop{\mathrm{Mor}}\nolimits (W, \text{cosk}_ n V) \\ & = & \mathop{\mathrm{Mor}}\nolimits (W, \text{cosk}_ n U \times \text{cosk}_ n V) \end{eqnarray*}
The lemma follows. $\square$
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