Hi During a category theory course given by Martin Hyland in 1995, the name "Parametrized Representability" was given to the following consequence of Yoneda. (I can't remember what was covariant or contravariant, but it's not important.) Let I and C be categories. Let F : I x C^op --> Set be a functor. Suppose that for each object i in I, the functor F(i,-) has a representing object i.e. an object Vi together with an isomorphism F(i,x) iso C(x,Vi) natural in x (1) Then V has a unique extension to a functor from I to C making (1) natural in i. This theorem is used all the time, e.g. to show that in a category with all binary products x is a functor, that in a cartesian category with all exponentials -> is a functor, etc. But I couldn't find a reference in Mac Lane. Does anyone know of a reference I could cite, preferably using the term "parametrized representability"? Thanks Paul 10-Jul-2002 09:53:42 -0300,1936;000000000000-00000012