Lemma 101.41.3. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. If f is separated, then f satisfies the uniqueness part of the valuative criterion.
Proof. Since f is separated we see that all categories of dotted arrows are setoids by Lemma 101.39.2. Consider a diagram
and a 2-morphism \gamma : y \circ j \to f \circ x as in Definition 101.39.1. Consider two objects (a, \alpha , \beta ) and (a', \beta ', \alpha ') of the category of dotted arrows. To finish the proof we have to show these objects are isomorphic. The isomorphism
means that (a, a', \beta ' \circ \beta ^{-1}) is a morphism \mathop{\mathrm{Spec}}(A) \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}. On the other hand, \alpha and \alpha ' define a 2-arrow
Here we use that both (a, \alpha , \beta ) and (a', \alpha ', \beta ') are dotted arrows with respect to \gamma . We obtain a commutative diagram
with 2-commutativity witnessed by (\alpha , \alpha '). Now \Delta _ f is representable by algebraic spaces (Lemma 101.3.3) and proper as f is separated. Hence by Lemma 101.39.13 and the valuative criterion for properness for algebraic spaces (Morphisms of Spaces, Lemma 67.44.1) we see that there exists a dotted arrow. Unwinding the construction, we see that this means (a, \alpha , \beta ) and (a', \alpha ', \beta ') are isomorphic in the category of dotted arrows as desired. \square
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