Perhaps "representably fully faithful" is a good name. In Cat, they are the fully faithful functors. In V-Cat they may not be fully faithful V-functors: see (with R.F.C. Walters) Yoneda structures on 2-categories, J. Algebra 50 (1978) 350-379 Also see (with A. Carboni, S. Johnson and D. Verity) Modulated bicategories, J. Pure Appl. Algebra 94 (1994) 229-282 ---Ross On 31/03/2007, at 3:44 AM, Robin Houston wrote:
A functor F: C -> D is full and faithful just when, for all categories X and functors G, H: X -> C, the whiskering action of F induces a bijection between [G, H] and [FG, FH] (where [G, H] denotes the set of natural transformations from G to H).
Clearly this formulation makes sense in any bicategory. Is there a name for 1-cells with this property?