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

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

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

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

## Comments (0)