Let $f$ be a morphism of algebraic stacks which is representable by algebraic spaces. In Properties of Stacks, Section 100.3 we have defined what it means for $f$ to be universally open. Here is another characterization.
Lemma 101.11.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces. The following are equivalent
$f$ is universally open (as in Properties of Stacks, Section 100.3), and
for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the morphism of topological spaces $|\mathcal{Z} \times _\mathcal {Y} \mathcal{X}| \to |\mathcal{Z}|$ is open.
Proof. Assume (1), and let $\mathcal{Z} \to \mathcal{Y}$ be as in (2). Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Z}$. By assumption the morphism $V \times _\mathcal {Y} \mathcal{X} \to V$ of algebraic spaces is universally open, in particular the map $|V \times _\mathcal {Y} \mathcal{X}| \to |V|$ is open. By Properties of Stacks, Section 100.4 in the commutative diagram
the horizontal arrows are open and surjective, and moreover
is surjective. Hence as the left vertical arrow is open it follows that the right vertical arrow is open. This proves (2). The implication (2) $\Rightarrow $ (1) follows from the definitions. $\square$
Thus we may use the following natural definition.
Definition 101.11.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
We say $f$ is open if the map of topological spaces $|\mathcal{X}| \to |\mathcal{Y}|$ is open.
We say $f$ is universally open if for every morphism of algebraic stacks $\mathcal{Z} \to \mathcal{Y}$ the morphism of topological spaces
is open, i.e., the base change $\mathcal{Z} \times _\mathcal {Y} \mathcal{X} \to \mathcal{Z}$ is open.
\[ |\mathcal{Z} \times _\mathcal {Y} \mathcal{X}| \to |\mathcal{Z}| \]
Lemma 101.11.3. The base change of a universally open morphism of algebraic stacks by any morphism of algebraic stacks is universally open.
Proof. This is immediate from the definition. $\square$
Lemma 101.11.4. The composition of a pair of (universally) open morphisms of algebraic stacks is (universally) open.
Proof. Omitted. $\square$
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