The Stacks project

Remark 6.32.5. Let $i : Z \to X$ be a closed immersion of topological spaces as above. Let $x \in X$, $x \not\in Z$. Let $\mathcal{F}$ be a sheaf of sets on $Z$. Then $(i_*\mathcal{F})_ x = \{ * \} $ by Lemma 6.32.1. Hence if $\mathcal{F} = * \amalg *$, where $*$ is the singleton sheaf, then $i_*\mathcal{F}_ x = \{ *\} \not= i_*(*)_ x \amalg i_*(*)_ x$ because the latter is a two point set. According to our conventions in Categories, Section 4.23 this means that the functor $i_*$ is not right exact as a functor between the categories of sheaves of sets. In particular, it cannot have a right adjoint, see Categories, Lemma 4.24.6.

On the other hand, we will see later (see Modules, Lemma 17.6.3) that $i_*$ on abelian sheaves is exact, and does have a right adjoint, namely the functor that associates to an abelian sheaf on $X$ the sheaf of sections supported in $Z$.