Dear Steve Sorry to cause you grief with the "fully faithful" terminology which is common in categorical papers, but perhaps not textbooks. The French use "pleinement fidele". The point about Cauchy-Morita completion is this: we cannot recapture a category A from its presheaf category P(A); we can only capture, up to equivalence, the Cauchy completion Q(A) of A. Given F : A --> B, if F* (or P(F) : P(B) --> P(A)) is fully faithful then the same will be true for Q(F) replacing F. For ordinary categories, Q(A) is the completion of A wrt splitting idempotents; ie, the full subcat of P(A) consisting of retracts of representables. Hence my point about F being surjective on objects up to retraction. [For additive categories, Q(A) is the full subcat of P(A) (additive ab-gp-valued presheaves) consisting of retracts of finite direct sums of representables.] +++++++++++++++++++++++++++++++++++++++++++++