The inertia of a stack in setoids is trivial

Full faithfulness in the setoid case. Suppose every fibre category of $\mathcal{X}$ is a setoid. Then the only objects of ${\mathcal{I}}_{\mathcal{X}}(T)$ are $(x,{\mathrm{i}\mathrm{d}}_x)$. For any two such objects, every isomorphism $\gamma \text{\hspace{0.17em}⁣}:x\to y$ in ${\mathcal{X}}_T$ satisfies $\gamma {\mathrm{i}\mathrm{d}}_x={\mathrm{i}\mathrm{d}}_y\gamma$, so it lifts uniquely to a morphism $(x,{\mathrm{i}\mathrm{d}}_x)\to (y,{\mathrm{i}\mathrm{d}}_y)$. Thus the projection is fully faithful on each fibre.

Essential surjectivity in the setoid case. For every $x\in {\mathcal{X}}_T$, the object $(x,{\mathrm{i}\mathrm{d}}_x)$ of ${\mathcal{I}}_{\mathcal{X}}(T)$ maps to $x$, so the projection is essentially surjective on every fibre. The functor $x\mapsto (x,{\mathrm{i}\mathrm{d}}_x)$, sending an arrow $\gamma$ to the same arrow $\gamma$, is an explicit inverse over the base: the inertia condition holds for identity automorphisms, and both composites are identities because every automorphism is the identity. Thus $\pi$ is an equivalence of stacks in groupoids, without using a choice-based converse to fibrewise essential surjectivity. Conversely, suppose $\pi$ is an equivalence. For any $x\in {\mathcal{X}}_T$ and $\alpha \in \mathrm{Aut}(x)$, full faithfulness applied to $(x,{\mathrm{i}\mathrm{d}}_x)$ and $(x,\alpha )$ lifts the identity $x\to x$ to a morphism between them. The inertia-morphism condition in [F1] then gives ${\mathrm{i}\mathrm{d}}_x=\alpha$, so every fibre category is a setoid.

The stack in setoids of an algebraic space. If $\mathcal{X}={\mathcal{S}}_Z$ then ${\mathcal{X}}_T$ is the discrete groupoid on ${\mathrm{Mor}}_S(T,Z)$ by [F2], so its only automorphisms are identities and step 1.2 shows that ${\mathcal{I}}_{{\mathcal{S}}_Z}\to {\mathcal{S}}_Z$ is an equivalence; the fibre category of ${\mathcal{I}}_{{\mathcal{S}}_Z}$ over $T$ is therefore the discrete groupoid on ${\mathrm{Mor}}_S(T,Z)$, which is exactly the fibre category of ${\mathcal{S}}_Z$.