Lemma 37.3.1. Let $(f, f') : (X \subset X') \to (S \subset S')$ be a morphism of thickenings. Then

$f$ is an affine morphism if and only if $f'$ is an affine morphism,

$f$ is a surjective morphism if and only if $f'$ is a surjective morphism,

$f$ is quasi-compact if and only if $f'$ quasi-compact,

$f$ is universally closed if and only if $f'$ is universally closed,

$f$ is integral if and only if $f'$ is integral,

$f$ is (quasi-)separated if and only if $f'$ is (quasi-)separated,

$f$ is universally injective if and only if $f'$ is universally injective,

$f$ is universally open if and only if $f'$ is universally open,

$f$ is quasi-affine if and only if $f'$ is quasi-affine, and

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