Lemma 8.11.8. Let $p : \mathcal{S} \to \mathcal{C}$ be a gerbe over a site $\mathcal{C}$. Assume that for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$ the sheaf of groups $\mathit{Aut}(x) = \mathit{Isom}(x, x)$ on $\mathcal{C}/U$ is abelian. Then there exist
a sheaf $\mathcal{G}$ of abelian groups on $\mathcal{C}$,
for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and every $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$ an isomorphism $\mathcal{G}|_ U \to \mathit{Aut}(x)$
such that for every $U$ and every morphism $\varphi : x \to y$ in $\mathcal{S}_ U$ the diagram
\[ \xymatrix{ \mathcal{G}|_ U \ar[d] \ar@{=}[rr] & & \mathcal{G}|_ U \ar[d] \\ \mathit{Aut}(x) \ar[rr]^{\alpha \mapsto \varphi \circ \alpha \circ \varphi ^{-1}} & & \mathit{Aut}(y) } \]
is commutative.
Proof.
Let $x, y$ be two objects of $\mathcal{S}$ with $U = p(x) = p(y)$.
If there is a morphism $\varphi : x \to y$ over $U$, then it is an isomorphism and then we indeed get an isomorphism $\mathit{Aut}(x) \to \mathit{Aut}(y)$ sending $\alpha $ to $\varphi \circ \alpha \circ \varphi ^{-1}$. Moreover, since we are assuming $\mathit{Aut}(x)$ is commutative, this isomorphism is independent of the choice of $\varphi $ by a simple computation: namely, if $\psi $ is a second such map, then
\[ \varphi \circ \alpha \circ \varphi ^{-1} = \psi \circ \psi ^{-1} \circ \varphi \circ \alpha \circ \varphi ^{-1} = \psi \circ \alpha \circ \psi ^{-1} \circ \varphi \circ \varphi ^{-1} = \psi \circ \alpha \circ \psi ^{-1} \]
The upshot is a canonical isomorphism of sheaves $\mathit{Aut}(x) \to \mathit{Aut}(y)$. Furthermore, if there is a third object $z$ and a morphism $y \to z$ (and hence also a morphism $x \to z$), then the canonical isomorphisms $\mathit{Aut}(x) \to \mathit{Aut}(y)$, $\mathit{Aut}(y) \to \mathit{Aut}(z)$, and $\mathit{Aut}(x) \to \mathit{Aut}(z)$ are compatible in the sense that
\[ \xymatrix{ \mathit{Aut}(x) \ar[rd] \ar[rr] & & \mathit{Aut}(z) \\ & \mathit{Aut}(y) \ar[ru] } \]
commutes.
If there is no morphism from $x$ to $y$ over $U$, then we can choose a covering $\{ U_ i \to U\} $ such that there exist morphisms $x|_{U_ i} \to y|_{U_ i}$. This gives canonical isomorphisms
\[ \mathit{Aut}(x)|_{U_ i} \longrightarrow \mathit{Aut}(y)|_{U_ i} \]
which agree over $U_ i \times _ U U_ j$ (by canonicity). By glueing of sheaves (Sites, Lemma 7.26.1) we get a unique isomorphism $\mathit{Aut}(x) \to \mathit{Aut}(y)$ whose restriction to any $U_ i$ is the canonical isomorphism of the previous paragraph. Similarly to the above these canonical isomorphisms satisfy a compatibility if we have a third object over $U$.
What if the fibre category of $\mathcal{S}$ over $U$ is empty? Well, in this case we can find a covering $\{ U_ i \to U\} $ and objects $x_ i$ of $\mathcal{S}$ over $U_ i$. Then we set $\mathcal{G}_ i = \mathit{Aut}(x_ i)$. By the above we obtain canonical isomorphisms
\[ \varphi _{ij} : \mathcal{G}_ i|_{U_ i \times _ U U_ j} \longrightarrow \mathcal{G}_ j|_{U_ i \times _ U U_ j} \]
whose restrictions to $U_ i \times _ U U_ j \times _ U U_ k$ satisfy the cocycle condition explained in Sites, Section 7.26. By Sites, Lemma 7.26.4 we obtain a sheaf $\mathcal{G}$ over $U$ whose restriction to $U_ i$ gives $\mathcal{G}_ i$ in a manner compatible with the glueing maps $\varphi _{ij}$.
If $\mathcal{C}$ has a final object $U$, then this finishes the proof as we can take $\mathcal{G}$ equal to the sheaf we just constructed. In the general case we need to verify that the sheaves $\mathcal{G}$ constructed over varying $U$ are compatible in a canonical manner. This is omitted.
$\square$
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