Mike Barr has pointed out that the proof in my last posting of LEMMA: The object I = H^R is injective in *F*. doesn't work. (It was actually the fourth proof I had come up with. I wondered why it was so much simpler). So here's one that does work (and is just about as simple). Let O | H^R | H^A --> H^B --> T --> O be exact (all vertical arrows point down). We seek a retraction for H^R --> T. Since H^R is projective (as is any representable) we may choose a map H^R --> H^B to yield a commutative triangle. The full subcategory of representables is closed under finite limits, so let H^C --> H^R | | H^A --> H^B be a pullback in *F* and let B --> A | | R --> C be the corresponding pushout in the category of f.p R-modules. The map from H^C to T is the zero map and we use the hypothesis that H^R --> T is monic to infer that H^C --> H^R, hence R --> C, are zero maps. Let O --> K --> B --> A be exact. It is an exercise in abelian categories that R --> C = 0 implies K --> B --> R is epi. Now (finally using the projectivity of R) choose a retraction R --> K. The map H^A --> H^B --> H^K --> H^R is of course, a zero map and we may factor H^B --> H^K --> H^R as H^B --> T --> H^R. The map T --> H^R is easily checked to be the retraction we seek.