defining grothendieck topology on category of etale schemes over S--cov(ET/S) consists of families like {X_i \to X} where X is the union of images by those morphisms now, given some assumptions like composition of etales is etale, and etale morphisms lift by fiber products, and gf & g etale imply f is etale, we see this category has fiber products but why is the lifted cover by a morphism again a surjective system, i.e., why is {X_i x_S V \to V} such that V = union of images? im thinking to check it locally, but it seems like a mess--hoping for a clean and formal way any ideas?