The first thing to note is that your question, as stated, is meaningless. The proper statement is that the induced functor has left and rights adjoints. If for one of them, one of the composites you mention is naturally equivalent to the identity, then another could be chosen (using AC) for which the composite in question was equal to the identity. On the other hand, in the usual set theory, the usual construction that shows that adjoint exists will use more complicated sets than the functor and thus the composite will never be the identity. Thus the only question that can meaningfully raised is when one or the other of the composites is naturally equivalent to the identity. I think sufficient conditions are known, although I don't recall them offhand, but it seems awfully unlikely to me that useful necessary and sufficient conditions are known. Michael Barr ==============================================================================