I know that the Torelli morphism $t\_g:\mathcal{M}\_g\rightarrow \mathcal{A}\_g$ between the stacks of smooth curves of genus $g$ and principally polarized abelian varieties of dimension $g$ is of order 2 ramified on the hyperelliptic locus. 
I also know that the natural map $\sigma:Aut(C)\rightarrow Aut (J(C),\Theta)$ which to an automorphism of the smooth curve $C$ associates the induced automorphism on $(J(C),\Theta)$ is an isomorphism for $C$ hyperelliptic and is such that $Aut(J(C),\Theta)=<Aut(C),-1>$ otherwise.


My question is: how are the two notions linked? I know this is a trivial question, but i can't see how, for a non-hyperelliptic curve $C$, $Aut(J(C),\Theta)=<Aut(C),-1>$ should imply that $t\_g^{-1}(J(C),\Theta)$ consists of two points in the stack $\mathcal{M}\_g$.. 



Map between stacks and automorphism groups
Look at the definition of fiber product in the category of stacks: For a scheme $S$, the $S$-points of the fiber product $X \times\_Z Y$ are triples $(x, y, \sigma)$ where $x \in X(S), y \in Y(S)$and $\sigma$ is a (iso)morphism between the images of $x$ and $y$ in the groupoid $Z(S)$.


Let $c:Spec\ k \to \mathcal{A}\_g$ be the morphism corresponding to the point $(J(C), \Theta)$. The $Spec\ k$ valued points of $\mathcal{M}\_g \times\_{t\_g, \mathcal{A}\_g, c} Spec\ k$ are triples $(Spec\ k \to \mathcal{M}\_g, Spec \ k \to Spec \ k, \sigma)$ where $\sigma$ is an isomorphism between images in $\mathcal{A}\_g$. So if the map 
$Spec\ k \to \mathcal{M}\_g$ corresponds to $C$ (which it must, for a $\sigma$ to exist, by Torelli theorem), then composing with $t\_g$ corresponds to $(J(C), \Theta)$ in $\mathcal{A}\_g$, and we can choose $\sigma$ to be an automorphism of $(J(C), \Theta)$.


