This is called the sieve generated by {"X""α" → "X"}. Now choose a topology. Say that {"X""α" → "X"} is a covering family if and only if the sieve that it generates is a covering sieve for the given topology. It is easy to check that this defines a pretopology. (PT 3) is sometimes replaced by a weaker axiom:   (PT 3) implies (PT 3'), but not conversely. However, suppose that we have a collection of covering families that satisfies (PT 0) through (PT 2) and (PT 3'), but not (PT 3). These families generate a pretopology. The topology generated by the original collection of covering families is then the same as the topology generated by the pretopology, because the sieve generated by an isomorphism "Y" → "X" is Hom(−, "X"). Consequently, if we restrict our attention to topologies, (PT 3) and (PT 3') are equivalent. Let "C" be a category and let "J" be a Grothendieck topology on "C". The pair ("C", "J") is called a site. A presheaf on a category is a contravariant functor from "C" to the category of all sets. Note that for this definition "C" is not required to have a topology. A sheaf on a site, however, should allow gluing, just like sheaves in classical topology. Consequently, we define a sheaf on a site to be a presheaf "F" such that for all objects "X" and all covering sieves "S" on "X", the natural map Hom(Hom(−, "X"), "F") → Hom("S", "F"), induced by the inclusion of "S" into Hom(−, "X"), is a bijection. Halfway in between a presheaf and a sheaf is the notion of a separated presheaf, where the natural map above is required to be only an injection, not a bijection, for all sieves "S". A morphism of presheaves or of sheaves is a natural transformation of functors. The category of all sheaves on "C" is the topos defined by the site ("C", "J"). Using the Yoneda lemma, it is possible to show that a presheaf on the category "O"("X") is a sheaf on the topology defined above if and only if it is a sheaf in the classical sense.