Regular coverage ($R$)

The regular coverage on a small regular category takes each regular epimorphism as a one-arrow cover. Pullback and composition stability make these a coverage basis. It is subcanonical because maps constant on the kernel pair of a regular epimorphism descend uniquely to its quotient.

If a family of subsheaves covers $yA$, local membership of $1_A$ gives a regular epimorphism $\alpha :B\to A$ belonging to one subsheaf. The sheaf condition descends that section to $1_A$ in the same subsheaf. All maps into $A$ are restrictions of $1_A$, so that subsheaf is $yA$. Thus representable sheaves for the regular coverage are irreducible even for arbitrary unions.

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 $F\prime$.