Lemma 37.61.9. Let $i : Z \to X$ and $j : Z \to Y$ be closed immersions of schemes. Let $f : X' \to X$ and $g : Y' \to Y$ be morphisms of schemes and let $\varphi : X' \times _{X, i} Z \to Y' \times _{Y, j} Z$ be an isomorphism of schemes over $Z$. Consider the morphism

Then we have

$h$ is locally of finite type if and only if $f$ and $g$ are locally of finite type,

$h$ is flat if and only if $f$ and $g$ are flat,

$h$ is flat and locally of finite presentation if and only if $f$ and $g$ are flat and locally of finite presentation,

$h$ is smooth if and only if $f$ and $g$ are smooth,

$h$ is étale if and only if $f$ and $g$ are étale, and

add more here as needed.

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