Lemma 102.14.4. Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemma 102.14.2.

There exists a functor

\[ g_! : \textit{Mod}(\mathcal{X}_{lisse,{\acute{e}tale}}, \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \longrightarrow \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_{\mathcal{X}}) \]which is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$ on abelian sheaves and $g^*g_! = \text{id}$.

There exists a functor

\[ g_! : \textit{Mod}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \longrightarrow \textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}}) \]which is left adjoint to $g^*$. Moreover it agrees with the functor $g_!$ on abelian sheaves and $g^*g_! = \text{id}$.

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