Lemma 18.4.2 (03A8)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.4: Free abelian presheaves
Lemma 18.4.2 (cite)

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Lemma 18.4.2. Let $\mathcal{C}$ be a category. Let $\mathcal{G}$ , $\mathcal{F}$ be a presheaves of sets. Let $\mathcal{A}$ be an abelian presheaf. Let $U$ be an object of $\mathcal{C}$ . Then we have

$\begin{align*} \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F}) & = \mathcal{F}(U), \\ \mathop{\mathrm{Mor}}\nolimits _{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_\mathcal {G}, \mathcal{A}) & = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{G}, \mathcal{A}), \\ \mathop{\mathrm{Mor}}\nolimits _{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_ U, \mathcal{A}) & = \mathcal{A}(U). \end{align*}$

All of these equalities are functorial.

Proof. Omitted. $\square$

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