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I'm struggling again with Joyal-Tierney's "Strong stacks & classifying spaces", and in the proof of the main theorem of the paper, where stacks are characterized as internal groupoids $\mathbb G$ satisfying a "weak lifting condition", appears a claim I'm not able to prove. Everything happens in a fixed Grothendieck topos.

Having a $\mathbb G$-torsor $E$ over an object $X$, Joyal builds a groupoid out of $E$, say $\mathbb E = (E_0=E,E_1,s,t,c)$, called its groupoid of elements, and claims that it is equivalent to the discrete internal category $\delta(X)$ on the object $X$. It seems a rather general fact that such maps $E\to X$ induce such an internal equivalence; $E\to X$ surely is the object part of the functor I'm looking for, but I'm not able to derive this result in any convincing way.

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