is typically given by the fiber product, producing an object over "t" from one over "s". the 'fiber' terminology is significant: the underlying heuristic is that "x" over "s" is a family of fibers, one for each 'point' of "s"; the fiber product is then the family on "t", which described by fibers is for each point of "t" the fiber at its image in "s". this set-theoretic language is too naïve to fit the required context, certainly, from algebraic geometry. it combines, though, with the use of the yoneda lemma to replace the 'point' idea with that of treating an object, such as "s", as 'as good as' the representable functor it sets up.