An étale presheaf on X is a contravariant functor from Ét(X) to the category of sets. A presheaf F is called an étale sheaf if it satisfies the analog of the usual gluing condition for sheaves on topological spaces. That is, F is an étale sheaf if and only if the following condition is true. Suppose that is an object of Ét(X) and that is a jointly surjective family of étale morphisms over X. For each i, choose a section xi of F over Ui. The projection map , which is loosely speaking the inclusion of the intersection of Ui and Uj in Ui, induces a restriction map . If for all i and j the restrictions of xi and xj to are equal, then there must exist a unique section x of F over U which restricts to xi for all i.
