Primitive decomposition of an atomic finite-surjection sheaf ($F\cong {\coprod }_C{F}_C$)

Partition the elements of a sheaf by their primitive-ancestor equivalence classes under bijection. Each class gives a sheaf subfunctor, because its membership is preserved and reflected along covering surjections. A representative primitive element names an epic representable map onto its class component. Thus every sheaf is the coproduct of these components, each an atom in a topos.