Lemma 101.6.3. A morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks is

a monomorphism if and only if $\Delta _{f, 1}$ is an isomorphism, and

representable by algebraic spaces if and only if $\Delta _{f, 1}$ is a monomorphism.

Moreover, the second diagonal $\Delta _{f, 2}$ is always a monomorphism.

**Proof.**
Recall from Properties of Stacks, Lemma 100.8.4 that a morphism of algebraic stacks is a monomorphism if and only if its diagonal is an isomorphism of stacks. Thus Lemma 101.6.2 can be rephrased as saying that a morphism is representable by algebraic spaces if the diagonal is a monomorphism. In particular, it shows that condition (3) of Lemma 101.3.4 is actually an if and only if, i.e., a morphism of algebraic stacks is representable by algebraic spaces if and only if its diagonal is a monomorphism.
$\square$

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