The question is about the last paragraph of Remark 12.4.3 in the book on algebraic stacks by Laumon and Moret-Bailly.

Let $S$ be a (quasi-separated) scheme and let $\mathscr{X}$ be an algebraic stack over $S$. Let $X \rightarrow \mathscr{X}$ be a smooth epimorphism from an algebraic space, and let $X_{\bullet} \rightarrow \mathscr{X}$ be the resulting simplicial hypercovering (the $n$-th layer is the $(n+1)$-fold fiber product of $X$ over $\mathscr{X}$). As explained in 12.4.4, there is a morphism of topoi $$(\epsilon^{-1}, \epsilon_*)\colon \mathscr{X}_{\text{lis-et}} \rightarrow (X_{\bullet})_{\text{et}}$$ from the lisse-etale topos of $\mathscr{X}$ to the etale topos of the simplicial algebraic space $X_{\bullet}$ with $\epsilon_*$ being just the "restriction" functor. The claim is that $\epsilon_*$ admits a right adjoint $\epsilon^!$ described as follows: for an open $U \rightarrow \mathscr{X}$ of the lisse-etale site and a sheaf $\mathscr{F}$ on $(X_{\bullet})_{\text{et}}$, let $X_{\bullet, U} \rightarrow U$ be the pulled back simplicial hypercovering, and let $(\mathscr{F}_{n, U})$ be the pulled back sheaf on $(X_{\bullet, U})_{\text{et}}$; then $$(\epsilon^!\mathscr{F})(U) := \mathrm{Eq}((P_{0, U})_*\mathscr{F}_{0, U} \Longrightarrow (P_{1, U})_*\mathscr{F}_{1, U})$$ (interpret the arrow as two parallel arrows), where $P_{i, U} \colon X_{\bullet, U} \rightarrow U$ is the projection, and the two arrows are coming from the two projections $X_{1, U} \rightarrow X_{0, U}$.

**My question is:** why is $\epsilon^!$ a right adjoint of $\epsilon_*$ as claimed in the book? How does one verify this? In particular, how does one define the adjunction map $\mathscr{G} \rightarrow \epsilon^!\epsilon_*(\mathscr{G})$ for a sheaf $\mathscr{G}$ on $\mathscr{X}_{\text{lis-et}}$? (The definition of this adjunction map seems to be the key point.)