In the Comparison between Functor Categories question from Bangor Sept 23, we had a functor F:C->D, but now instead of Set, consider an arbitrary category A and the induced functor A^F:A^D->A^C. Left and right adjoints, when they exist, are patched together from Kan extensions. For any particular T in A^C, we could have Ran_F(T), Lan_F(T) in A^D with n.t.(S,Ran_F(T)) iso n.t.(A^F(S),T) or n.t(Lan_F(T),S) iso n.t.(T,A^F(S)) which occurrences we call right or left Kan extensions of T along F. These are approximations of the variable object T by images of A^F, which I think of heuristically as a sort of Dedekind cut situation, in my own naive way, as follows: think of those S in A^D for which SF<=T and call the Lower also think of those S for which T<=SF and call them Upper here SF=A^F(S) of course, and X<=Y just means there is some n.t. X->Y. Then A^F images of Lowers approximate T from below, and Uppers from above. Of course we proceed to look for best approximations. Among Lowers, the closest image to T (w.r.t. <=) would be the sup, and among Uppers it's the inf, corresponding to pointwise Kan ext. as limits. So T gets caught in a squeeze play (like a real number), but still might have lots of room to bounce around. Back to the Bangor question (as least partly), we could ask what happened if T got hit on the nose by A^F with some Z in A^D? ie. ZF->T is identity n.t. Surely Z is then the best approx. to T from above and below since it is bang on. But does Z have to be (iso to) a Kan ext. of T along F? (which it needs to be if it is going to participate in any adjoint for A^F). Consider left extensions (inf of Uppers): from ZF=T<=SF we need to produce a n.t. Z->S. Again heuristically, the temptation is to cross off F on the right in a kind of epi maneuver, which relates to Steve Vickers' comment for rings. By the way, the very nice perspective on modules SJV mentioned also appears as Exercise 3(b) Chap VII p.415 in Mac Lane/Moerdijk '92 "Sheaves..". But epi for F seems too strong to be necessary, whereas the nice suggestion of Ross Street that each object of D should be a retract of an object in the image of F with F factoring in a certain friendly way looks really neat. Still, not clear it should be necessary. Anyway, might help to think about it in different ways. Cheers ........................................Al Vilcius, Toronto ==============================================================================