The Stacks project

Proof. In both cases, the existence of the functor $g_!$ follows from Modules on Sites, Lemma 18.41.1. To see that $g_!$ agrees with the functor on abelian sheaves we will show the maps Modules on Sites, Equation (18.41.2.1) are isomorphisms.

Lisse-étale case. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{lisse,{\acute{e}tale}})$ lying over a scheme $U$ with $x : U \to \mathcal{X}$ smooth. Consider the induced fully faithful functor

The right hand side is identified with $(\mathit{Sch}/U)_{\acute{e}tale}$ and the left hand side with the full subcategory of schemes $U'/U$ such that the composition $U' \to U \to \mathcal{X}$ is smooth. Thus Étale Cohomology, Lemma 59.49.2 applies.

Flat-fppf case. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{flat,fppf})$ lying over a scheme $U$ with $x : U \to \mathcal{X}$ flat. Consider the induced fully faithful functor

The right hand side is identified with $(\mathit{Sch}/U)_{fppf}$ and the left hand side with the full subcategory of schemes $U'/U$ such that the composition $U' \to U \to \mathcal{X}$ is flat. Thus Étale Cohomology, Lemma 59.49.2 applies.

In both cases the equality $g^*g_! = \text{id}$ follows from $g^* = g^{-1}$ and the equality for abelian sheaves in Lemma 103.14.2. $\square$