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.
$\square$

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