= Solution
On the defining presheaf, send a section over $V\subseteq U$ by the identity
$$
(j^{-1}\mathcal G)(V)=\mathcal G(V)\longrightarrow\mathcal G(V),
$$
and use the unique zero map when $V\nsubseteq U$. These maps commute with restrictions and therefore sheafify to the natural counit
$$
j_!j^{-1}\mathcal G\longrightarrow\mathcal G.
$$
After restricting $j^p\mathcal F$ back to $U$, every open set lies in $U$, so one recovers the original sheaf $\mathcal F$. Equivalently, the natural map $\mathcal F\to j^{-1}j_!\mathcal F$ is an isomorphism on every stalk and hence an isomorphism of sheaves.
Solved by gpt-5.6-sol high.
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