Corollaire 3.13 in "Champs algebriques" says that the diagonal 1-morphism of stacks $\Delta:\mathcal{X} \to \mathcal{X} \times\_S \mathcal{X}$ is representable if and only if the sheaf $\mathcal{Isom}(x,y):\mathrm{Aff}/U \to \mathrm{Ens}$ is represented by an algebraic $U$-space, for every $U \in \mathrm{Aff}/S$ and all $x,y \in \mathcal{X}\_U$. 


I've seen this shown by claiming that the stack associated to $\mathcal{Isom}(x,y)$ is canonically isomorphic to $U \times\_{(x,y),\mathcal{X} \times\_S \mathcal{X}, \Delta} \mathcal{X}$. My question is how does one make this identification?



Representability of the diagonal morphism of stacks
This follows immediately from the definitions, in particular from the fiber product of stacks (see loc. cit. (2.2.2)). For $V \in \mathrm{Aff}$ (everything over the base $S$), a $V$-point of $U \times\_{X \times X} X$ is a triple $(i,z,\alpha)$, where $i$ is a $V$-point of $U$, $z$ is a $V$-point of $X$ and $\alpha : (i^\* x,i^\* y) \to (i^\*z ,i^\* z)$ is an isomorphism of $V$-points of $X \times X$. This comes down to an isomorphism $\alpha\_2^{-1} \circ \alpha\_1 : i^\* x \to i^\* y$, i.e. a $V$-point of $\mathrm{Isom}(x,y)$.


