Rigid coverage

A coverage on a category is rigid when each object is covered by the sieve generated by arrows from J-irreducible objects of a site. Restriction to their full subcategory identifies the sheaves on a site with its presheaf category. A presheaf on that subcategory extends by the right Kan extension $C\mapsto \mathrm{Nat}(yC{|}_{\mathcal{D}},F)$: every covering sieve restricts to the full representable on $\mathcal{D}$, which proves the sheaf condition directly.