 Solution

On the defining presheaf, send a section over $V\subseteq U$ by the identityand use the unique zero map when $V⊈U$. These maps commute with restrictions and therefore sheafify to the natural counit

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.