The Stacks project

Remark 12.29.4. Let $\mathcal{A}$, $\mathcal{B}$, $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be as In Lemma 12.29.3. In the presence of conditions (1) and (2) condition (4) is equivalent to $v$ being faithful. Moreover, condition (4) is needed. An example is to consider the case where the functors $u$ and $v$ are both the zero functor.