The Stacks project

Remark 17.6.4. In Sheaves, Remark 6.32.5 we showed that $i_*$ as a functor on the categories of sheaves of sets does not have a right adjoint simply because it is not exact. However, it is very close to being true, in fact, the functor $i_*$ is exact on sheaves of pointed sets, sections with support in $Z$ can be defined for sheaves of pointed sets, and $\mathcal{H}_ Z$ makes sense and is a right adjoint to $i_*$.