There is a need to say a few more elementary things on this question. My initial reply was short because I was busy teaching an intensive on-campus distance course, and I hoped someone else would respond in more detail. The equality question is meaningless. Also, the term "natural equivalence" is a term used "classically" by category theorists, but not a very good one; I suggest we should try to avoid it since "equivalence" means something else. Let F : C --> D be a functor and let L --| F* --| R as in the question. Let e : L F* --> 1, n : 1 --> F* L, e' : F* R --> 1, n' : 1 --> R F* be the counits and units for the adjunctions. Fact 1: n invertible iff L fully faithful iff R fully faithful iff e' invertible. Fact 2: e invertible iff F* fully faithful iff n' invertible. So there are really only two problems (as I just notice Dusko Pavlovic has pointed out): (a) when is L fully faithful? (b) when is F* fully faithful? Answer to (a): L fully faithful iff F is fully faithful. Proof: Since we are taking Kan extensions along F of functors into Set, there are formulas for L (pointwise left Kan extension). Any formula can be used to show F fully faithful implies n invertible, so L is fully faithful. This must be in all the textbooks (eg Mac Lane). Conversely, there is a square which commutes up to isomorphism (or equality if we choose L suitably on representables) involving L, two Yoneda embeddings, and F^op : C^op --> D^op. If L is fully faithful, so is F^op (since the Yon embs are), and so, so is F.///// Now some comments on (b): (i) Let's call F a "localization" when there exists a set S of arrows in C for which F is the universal functor out of C inverting the arrows of S. Given F, if there is an S, the set of arrows inverted by F is the largest such S. Localizations are bijective on objects. Localizations are coinverters of natural transformations between functors into C. If F is a localization, it follows that F* is an inverter of some natural transformation and hence is fully faithful. (ii) F* fully faithful does NOT imply F localization. For, if F induces an equivalence of categories on the cauchy (idempotent splitting) completions of C, D it will still have F* fully faithful, but need not have F bijective on objects. (iii) F* conservative (= reflects isos) iff e epic iff each object d of D is a retract of an object Fc for c in C. (iv) These things suggest to me that the answer to (b) could be: "F* fully faithful iff each object of D is a retract of an object in the image of F, and F = G H with G fully faithful and H a localization". However, localizations in Cat are notoriously difficult to characterize (easier in Lex, Rex, Pb, . . .). The above goes over to enriched categories with appropriate change in the notion of cauchy completion (and epic in (iii) becomes extremal epic); eg, additive categories for the context of Steve Vickers' response. Regards, Ross ==============================================================================