Lemma 7.8.5 (0G1K)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 7: Sites and Sheaves
Section 7.8: Families of morphisms with fixed target
Lemma 7.8.5 (cite)

Lemma 7.8.5. Let $\mathcal{C}$ be a category. Let $\mathcal{V} = \{ V_ j \to U\} _{j \in J} \to \mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a morphism of families of maps with fixed target of $\mathcal{C}$ given by $\text{id} : U \to U$, $\alpha : J \to I$ and $f_ j : V_ j \to U_{\alpha (j)}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{C}$. If $\mathcal{F}(U) \to \prod _{j \in J} \mathcal{F}(V_ j)$ is injective then $\mathcal{F}(U) \to \prod _{i \in I} \mathcal{F}(U_ i)$ is injective.

Proof. Omitted. $\square$

Comments (2)

Comment #7475 by Dun Liang on July 12, 2022 at 03:30

Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I} \to \mathcal{V} = \{ V_ j \to U\} _{j \in J}$ be a morphism of families of maps with fixed target of $\mathcal C$ given by ...

should this be Let $\mathcal{V} = \{ V_ j \to U\} _{j \in J}\to \mathcal{U} = \{ U_i\to U\} _{i\in I}$ be a morphism of families of maps with fixed target of $\mathcal C$ given by ...

Comment #7623 by Stacks Project on August 14, 2022 at 10:38

Thanks. Changed it here.

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