Lemma 14.13.3. Let $\mathcal{C}$ be a category such that the coproduct of any two objects of $\mathcal{C}$ exists. Let us temporarily denote $\textit{FSSets}$ the category of simplicial sets all of whose components are finite nonempty.

1. The rule $(U, V) \mapsto U \times V$ defines a functor $\textit{FSSets} \times \text{Simp}(\mathcal{C}) \to \text{Simp}(\mathcal{C})$.

2. For every $U$, $V$ as above there is a canonical map of simplicial objects

$U \times V \longrightarrow V$

defined by taking the identity on each component of $(U \times V)_ n = \coprod _ u V_ n$.

Proof. Omitted. $\square$

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