Lemma 7.22.1 (00XX)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 7: Sites and Sheaves
Section 7.22: Cocontinuous functors which have a right adjoint
Lemma 7.22.1 (cite)

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Lemma 7.22.1. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$, and $v : \mathcal{D} \to \mathcal{C}$ be functors. Assume that $u$ is cocontinuous and that $v$ is a right adjoint to $u$. Let $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be the morphism of topoi associated to $u$, see Lemma 7.21.1. Then

for a sheaf $\mathcal{F}$ on $\mathcal{C}$ the sheaf $g_*\mathcal{F}$ is equal to the presheaf $v^ p\mathcal{F}$, in other words, $(g_*\mathcal{F})(V) = \mathcal{F}(v(V))$, and
for a sheaf $\mathcal{G}$ on $\mathcal{D}$ we have $g^{-1}\mathcal{G} = (v_ p\mathcal{G})^\# $.

Proof. For $\mathcal{F}$ as in (1) we have

\[ g_*\mathcal{F} = {}_ su\mathcal{F} = {}_ pu\mathcal{F} = v^ p\mathcal{F} = \mathcal{F} \circ v \]

The first equality is Lemma 7.21.1. The second equality is Lemma 7.20.2. The third equality is Lemma 7.19.3. The final equality is the definition of $v^ p$ in Section 7.5. This proves (1). For $\mathcal{G}$ as in (2) we have

\[ g^{-1}\mathcal{G} = (u^ p\mathcal{G})^\# = (v_ p\mathcal{G})^\# \]

The first equality is Lemma 7.21.1. The second equality is Lemma 7.19.3. $\square$

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