Lemma 4.24.4. Let $u$ be a left adjoint to $v$ as in Definition 4.24.1. Then

1. $u$ is fully faithful $\Leftrightarrow$ $\text{id} \cong v \circ u$ $\Leftrightarrow$ $\eta : \text{id} \to v \circ u$ is an isomorphism,

2. $v$ is fully faithful $\Leftrightarrow$ $u \circ v \cong \text{id}$ $\Leftrightarrow$ $\epsilon : u \circ v \to \text{id}$ is an isomorphism.

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). $\square$

Comment #5571 by Tobias Schmidt on

At the beginning of line -2 of the proof of part (1) there should be no letter u, since the expression is supposed to be a preimage under u.

Comment #5575 by Jeroen Hekking on

I think there should actually be a $vu$ instead of only a $u$.

Comment #5752 by on

OK, I think you were both correct. But I was having trouble verifying the formular because I needed to prove that $\epsilon_{uvu(X)} = uv(\epsilon_{u(X)})$ which I failed to do (and I am not sure holds in general; it'd be mildly interesting to know if it is or not; a posteriori I think it holds in the setting it was being used in the proof of the lemma). Anyway, the lemma is fine as stated. See changes in this commit.

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