1. Under which precise conditions on F is L F^* = Id. 2. Under which precise conditions on F is R F^* = Id. 3. Under which precise conditions on F is F^* R = Id. 4. Under which precise conditions on F is F^* L = Id.
(I guess some people know the precise answers at once, but don't bother to tell us. Here is what I figured.) Suppose, as M.Barr and R.Street promptly suggested, that the equalities in the query denote the natural isos. Of course, * 1 and 2 are equivalent: they mean that F^* is fully faithful; and * 3 and 4 are equivalent: they say that the essential morphism of toposes R: (D,Set)-->(C,Set) is an injection. A sufficient condition for 1 and 2 is that F:C-->D is a stably initial functor. (Initial functors are orthogonal to discrete opfibrations: cf. e.g. "The comprehensive factorisation of a functor" by Street and Walters, early seventies. Now F^* is pulling back of discrete opfibrations along F. Use orthogonality to show that it is fully faithful.) A sufficient condition for 3 and 4 is that F is fully faithful. (Just write down the pointwise Kan-extension formula for L (or R): to calculate F^*L(G) at X from C, we use the diagram obtained by projecting the comma category F/FX to C; we apply G on this diagram and calculate the colimit. When F is fully faithful, this diagram is just a cocone to X - so the colimit is GX.) Now what are the necessary conditions? Best regards, Dusko Pavlovic ==============================================================================