Definition 100.14.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. We say $f$ is *universally injective* if for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the map

\[ |\mathcal{Z} \times _\mathcal {Y} \mathcal{X}| \to |\mathcal{Z}| \]

is injective.

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