Proof.
Proof of (1). Assume u is fully faithful. We will show \eta _ X : X \to v(u(X)) is an isomorphism. Let X' \to v(u(X)) be any morphism. By adjointness this corresponds to a morphism u(X') \to u(X). By fully faithfulness of u this corresponds to a unique morphism X' \to X. Thus we see that post-composing by \eta _ X defines a bijection \mathop{\mathrm{Mor}}\nolimits (X', X) \to \mathop{\mathrm{Mor}}\nolimits (X', v(u(X))). Hence \eta _ X is an isomorphism. If there exists an isomorphism \text{id} \cong v \circ u of functors, then v \circ u is fully faithful. By Lemma 4.24.3 we see that u is fully faithful. By the above this implies \eta is an isomorphism. Thus all 3 conditions are equivalent (and these conditions are also equivalent to v \circ u being fully faithful).
Part (2) is dual to part (1).
\square
Comments (3)
Comment #5571 by Tobias Schmidt on
Comment #5575 by Jeroen Hekking on
Comment #5752 by Johan on