The Stacks project

Lemma 7.27.3. Let $\mathcal{C}$ be a site. Let $U \to V$ be a morphism of $\mathcal{C}$. Assume $\mathcal{C}$ has fibre products. Let $j$ be as in Lemma 7.25.8. Then

  1. the functor $j : \mathcal{C}/U \to \mathcal{C}/V$ has a continuous right adjoint, namely the functor $v : (X/V) \mapsto (X \times _ V U/U)$,

  2. the functor $v$ defines a morphism of sites $\mathcal{C}/U \to \mathcal{C}/V$ whose associated morphism of topoi equals $j$, and

  3. we have $j_*\mathcal{F}(X/V) = \mathcal{F}(X \times _ V U/U)$.

Proof. Follows from Lemma 7.27.2 since $j$ may be viewed as a localization functor by Lemma 7.25.8. $\square$

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