The Stacks project

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). $\square$