Local membership closure for the regular coverage ($\overline{F\prime }(A)=\{x:\exists \alpha :B\twoheadrightarrow A,F(\alpha )x\in F\prime (B)\}$)

For a subfunctor $F\prime \subseteq F$ of a regular-coverage sheaf, its closure consists of sections whose restriction belongs to $F\prime$ along one covering regular epimorphism. Pullbacks prove that local membership is a subfunctor; composition of witnessing covers proves the sheaf condition. It is the smallest subsheaf containing $F\prime$.