If S is a sieve on X, and f: Y → X is a morphism, then left composition by f gives a sieve on Y called the pullback of S along f, denoted by fS. It is defined as the fibered product S ×Hom(−, X) Hom(−, Y) together with its natural embedding in Hom(−, Y). More concretely, for each object Z of C, fS(Z) = { g: Z → Y | fg S(Z) }, and fS inherits its action on morphisms by being a subfunctor of Hom(−, Y). In the classical example, the pullback of a collection {Vi} of subsets of U along an inclusion W → U is the collection {Vi∩W}.
If S is a sieve on X, and f Y → X is a morphism, then left composition by f gives a sieve on Y called the pullback of S along f, denoted by fS. It is defin,ed as the fibered product S ×Hom(− X) Hom(−, Y) together with its natural embedding in Hom(−, Y). More concretely, for each object Z of C, fS(Z) = { g: Z → Y | fg S(Z) } and fS inherits its action on morphisms by being a subfunct:or of Hom(−, Y). In the classical example, the pullback of a collection {Vi} of subsets of U along an inclusion W → U is the collection {Vi∩W}.