Lemma 14.13.2 (017D)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 14: Simplicial Methods
Section 14.13: Products with simplicial sets
Lemma 14.13.2 (cite)

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Lemma 14.13.2. Let $\mathcal{C}$ be a category such that the coproduct of any two objects of $\mathcal{C}$ exists. Let $U$ be a simplicial set. Let $V$ be a simplicial object of $\mathcal{C}$ . Assume that each $U_ n$ is finite nonempty. The functor $W \mapsto \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}(\mathcal{C})}(U \times V, W)$ is canonically isomorphic to the functor which maps $W$ to the set in Equation (14.13.0.1).

Proof. Omitted. $\square$

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