Lemma 96.17.6. Let f : \mathcal{X} \to \mathcal{Y} be a 1-morphism of categories fibred in groupoids over (\mathit{Sch}/S)_{fppf}. Let \tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} . The functor f^* : \textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {Y}) \to \textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X}) has a left adjoint f_! : \textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {Y}) which agrees with the functor f_! of Lemma 96.17.5 on underlying abelian sheaves. If f is faithful and \mathcal{X} has equalizers, then
f_! is exact, and
f^{-1}\mathcal{I} is injective in \textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X}) for \mathcal{I} injective in \textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {X}).
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