Terminology

Dear all, I'm sure the following question has been answered to. Could anyone give me a precise answer and references to this answer. Many thanks. QUESTION Let p: S --> X be a functor. What conditions should satisfy p to be called a structure functor, i.e. such that every object s of S can be thought of as a structure on the object p(s). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]