The Stacks project

Proof. Assume $\mathcal{S}_1$ is a gerbe over $\mathcal{C}$. By Lemma 8.5.4 we see $\mathcal{S}_2$ is a stack in groupoids over $\mathcal{C}$. Let $F : \mathcal{S}_1 \to \mathcal{S}_2$, $G : \mathcal{S}_2 \to \mathcal{S}_1$ be equivalences of categories over $\mathcal{C}$. Given $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we see that there exists a covering $\{ U_ i \to U\} $ such that $(\mathcal{S}_1)_{U_ i}$ is nonempty. Applying $F$ we see that $(\mathcal{S}_2)_{U_ i}$ is nonempty. Given $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $x, y \in \mathop{\mathrm{Ob}}\nolimits ((\mathcal{S}_2)_ U)$ there exists a covering $\{ U_ i \to U\} $ in $\mathcal{C}$ such that $G(x)|_{U_ i} \cong G(y)|_{U_ i}$ in $(\mathcal{S}_1)_{U_ i}$. By Categories, Lemma 4.35.9 this implies $x|_{U_ i} \cong y|_{U_ i}$ in $(\mathcal{S}_2)_{U_ i}$. $\square$