A cosieve under an object in is a family of morphisms in with domain closed under postcomposition with any morphism in . In other words, implies whenever the composite exists. Cosieves under are also said to be cosieve in the under category ; all such cosieves for varying are said to be cosieves on . Cosieves may be viewed as subfunctors of the (co)representable (covariant) functors .
Cosieves on may be organized into a category . For convenience we will note the domain of a sieve as a part of the data. Thus objects of are pairs of the form where and is a cosieve in . A morphism is a map such that the cosieve is a subset of . The usual composition of underlying morphisms in defines a composition in , because where . Note that .