Re: terminology in definitions of limits

I have always used the phrase "test object" in a slightly different sense. Namely, to refer to a tractably small collection of objects that one may use, not only to detect, but also to calculate some right adjoint. Thus in Set, one may take the terminal object; in Set/X, the elements 1-->X; in Cat, the ordinals 1, 2 and 3; in presheaf categories, the representables; and so on. The best case is that these test objects are colimit dense, since then your calculations always yield a right adjoint as soon as the functor you start with preserves colimits. Richard