Lemma 18.16.6. Let \mathcal{C} and \mathcal{D} be sites. Let g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) be the morphism of topoi associated to a continuous and cocontinuous functor u : \mathcal{C} \to \mathcal{D}.
If u has a left adjoint w, then g_! agrees with g_!^{\mathop{\mathit{Sh}}\nolimits } on underlying sheaves of sets and g_! is exact.
If in addition w is cocontinuous, then g_! = h^{-1} and g^{-1} = h_* where h : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) is the morphism of topoi associated to w.
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