= Local membership closure for the regular coverage
{title2=$\overline{F\prime}(A)=\{x:\exists\alpha:B\twoheadrightarrow A,\ F(\alpha)x\in F\prime(B)\}$}
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 $F\prime$.
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