The Stacks project

Proof. Let $f : X \to Y$ be an ind-quasi-affine morphism. For checking that every base change of $f$ is ind-quasi-affine, by Lemma 37.66.2, we may work Zariski locally on $Y$, so we assume that $Y$ is affine. Furthermore, we may also assume that in the base change morphism $Z \to Y$ the scheme $Z$ is affine, too. The base change $X \times _ Y Z \to X$ is an affine morphism, so, by Lemmas 37.66.3 and 37.66.4, the map $X \times _ Y Z \to Y$ is ind-quasi-affine. Then, by Lemma 37.66.5, the base change $X \times _ Y Z \to Z$ is ind-quasi-affine, as desired. $\square$