The Stacks project

Lemma 7.27.2. Let $\mathcal{C}$ be a site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Assume $\mathcal{C}$ has products of pairs of objects. Then

  1. the functor $j_ U$ has a continuous right adjoint, namely the functor $v(X) = X \times U / U$,

  2. the functor $v$ defines a morphism of sites $\mathcal{C}/U \to \mathcal{C}$ whose associated morphism of topoi equals $j_ U : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, and

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

Proof. The functor $v$ being right adjoint to $j_ U$ means that given $Y/U$ and $X$ we have

\[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(Y, X) = \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}/U}(Y/U, X \times U/U) \]

which is clear. To check that $v$ is continuous let $\{ X_ i \to X\} $ be a covering of $\mathcal{C}$. By the third axiom of a site (Definition 7.6.2) we see that

\[ \{ X_ i \times _ X (X \times U) \to X \times _ X (X \times U)\} = \{ X_ i \times U \to X \times U\} \]

is a covering of $\mathcal{C}$ also. Hence $v$ is continuous. The other statements of the lemma follow from Lemmas 7.22.1 and 7.22.2. $\square$

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