The Stacks project

Remark 59.48.2. In the situation of Lemma 59.48.1 it is true that the canonical map $\mathcal{F} \to f_{big}^{-1}f_{big!}\mathcal{F}$ is an isomorphism for any sheaf of sets $\mathcal{F}$ on $(\mathit{Sch}/X)_\tau $. The proof is the same. This also holds for sheaves of abelian groups. However, note that the functor $f_{big!}$ for sheaves of abelian groups is defined in Modules on Sites, Section 18.16 and is in general different from $f_{big!}$ on sheaves of sets. The result for sheaves of abelian groups follows from Modules on Sites, Lemma 18.16.4.