Lemma 100.4.7. Let $f : \mathcal{X} \to \mathcal{Z}$, $g : \mathcal{Y} \to \mathcal{Z}$ and $\mathcal{Z} \to \mathcal{T}$ be morphisms of algebraic stacks. Consider the induced morphism $i : \mathcal{X} \times _\mathcal {Z} \mathcal{Y} \to \mathcal{X} \times _\mathcal {T} \mathcal{Y}$. Then

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

if $\Delta _{\mathcal{Z}/\mathcal{T}}$ is quasi-separated, then $i$ is quasi-separated,

if $\Delta _{\mathcal{Z}/\mathcal{T}}$ is separated, then $i$ is separated,

if $\mathcal{Z} \to \mathcal{T}$ is DM, then $i$ is unramified,

if $\mathcal{Z} \to \mathcal{T}$ is quasi-DM, then $i$ is locally quasi-finite,

if $\mathcal{Z} \to \mathcal{T}$ is separated, then $i$ is proper, and

if $\mathcal{Z} \to \mathcal{T}$ is quasi-separated, then $i$ is quasi-compact and quasi-separated.

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