The Stacks project

Lemma 101.5.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $Z$ be an algebraic space and let $x_ i : Z \to \mathcal{X}$, $i = 1, 2$ be morphisms. Then

1. $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2)$ is a group algebraic space over $Z$,

2. there is an exact sequence of groups

   $$0 \to \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2) \to \mathit{Isom}_\mathcal {X}(x_2, x_2) \to \mathit{Isom}_\mathcal {Y}(f \circ x_2, f \circ x_2)$$

3. there is a map of algebraic spaces $\mathit{Isom}_\mathcal {X}(x_1, x_2) \to \mathit{Isom}_\mathcal {Y}(f \circ x_1, f \circ x_2)$ such that for any $2$-morphism $\alpha : f \circ x_1 \to f \circ x_2$ we obtain a cartesian diagram

   $$\xymatrix{ \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha (x_1, x_2) \ar[d] \ar[r] & Z \ar[d]^\alpha \\ \mathit{Isom}_\mathcal {X}(x_1, x_2) \ar[r] & \mathit{Isom}_\mathcal {Y}(f \circ x_1, f \circ x_2) }$$

4. for any $2$-morphism $\alpha : f \circ x_1 \to f \circ x_2$ the algebraic space $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha (x_1, x_2)$ is a pseudo torsor for $\mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x_2, x_2)$ over $Z$.