Lemma 7.5.6 (04D2)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 7: Sites and Sheaves
Section 7.5: Functoriality of categories of presheaves
Lemma 7.5.6 (cite)

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Lemma 7.5.6. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories. For any object $U$ of $\mathcal{C}$ we have $u_ ph_ U = h_{u(U)}$ .

Proof. By adjointness of $u_ p$ and $u^ p$ we have

$\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{D})}(u_ ph_ U, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, u^ p\mathcal{G}) = u^ p\mathcal{G}(U) = \mathcal{G}(u(U))$

and hence by Yoneda's lemma we see that $u_ ph_ U = h_{u(U)}$ as presheaves. $\square$

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