Lemma 100.4.9. Let $f : \mathcal{X} \to \mathcal{T}$ be a morphism of algebraic stacks. Let $s : \mathcal{T} \to \mathcal{X}$ be a morphism such that $f \circ s$ is $2$-isomorphic to $\text{id}_\mathcal {T}$. Then

$s$ is representable by algebraic spaces and locally of finite type,

if $f$ is DM, then $s$ is unramified,

if $f$ is quasi-DM, then $s$ is locally quasi-finite,

if $f$ is separated, then $s$ is proper, and

if $f$ is quasi-separated, then $s$ is quasi-compact and quasi-separated.

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