Definition 14.17.1. Let $\mathcal{C}$ be a category such that the coproduct of any two objects exists. Let $U$ be a simplicial set, with $U_ n$ finite nonempty for all $n \geq 0$. Let $V$ be a simplicial object of $\mathcal{C}$. We denote $\mathop{\mathrm{Hom}}\nolimits (U, V)$ any simplicial object of $\mathcal{C}$ such that
\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}(\mathcal{C})}(W, \mathop{\mathrm{Hom}}\nolimits (U, V)) = \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}(\mathcal{C})}(W \times U, V) \]
functorially in the simplicial object $W$ of $\mathcal{C}$.
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