Lemma 8.11.3. Let $\mathcal{C}$ be a site. Let $p : \mathcal{X} \to \mathcal{C}$ and $q : \mathcal{Y} \to \mathcal{C}$ be stacks in groupoids. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories over $\mathcal{C}$. The following are equivalent

For some (equivalently any) factorization $F = F' \circ a$ where $a : \mathcal{X} \to \mathcal{X}'$ is an equivalence of categories over $\mathcal{C}$ and $F'$ is fibred in groupoids, the map $F' : \mathcal{X}' \to \mathcal{Y}$ is a gerbe (with the topology on $\mathcal{Y}$ inherited from $\mathcal{C}$).

The following two conditions are satisfied

for $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y})$ lying over $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a covering $\{ U_ i \to U\} $ in $\mathcal{C}$ and objects $x_ i$ of $\mathcal{X}$ over $U_ i$ such that $F(x_ i) \cong y|_{U_ i}$ in $\mathcal{Y}_{U_ i}$, and

for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $x, x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ U)$, and $b : F(x) \to F(x')$ in $\mathcal{Y}_ U$ there exists a covering $\{ U_ i \to U\} $ in $\mathcal{C}$ and morphisms $a_ i : x|_{U_ i} \to x'|_{U_ i}$ in $\mathcal{X}_{U_ i}$ with $F(a_ i) = b|_{U_ i}$.

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