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"?

See page 48 of G.M. Kelly "Basic Concepts of Enriched Category Theory" (Cambridge, 1982). This does it for enriched categories, including ordinary ones. (Max taught it too: in an honours course back in 1965.) Day and I [Monoidal bicategories and Hopf algebroids, Advances 129 (1997)] do it for bicategories (and use the word "parametrized") on page 107; and moreover, on page 114, we do it internal to an autonomous (= rigid = compact) monoidal bicategory. Ross 10-Jul-2002 13:09:18 -0300,6960;000000000000-00000015