The Stacks project

Proof. (Finite type morphisms of formal algebraic spaces are discussed in Formal Spaces, Section 87.24.) Namely, suppose that $Z \to X$ is a morphism from a scheme into $X$ such that $|Z|$ maps into $T$. From the cartesian square above we see that $Z \times _ X X'$ is an algebraic space representing $Z \times _{X_{/T}} X'_{/T'}$. Since $Z \times _ X X' \to Z$ is locally of finite type by Morphisms of Spaces, Lemma 67.23.3 we conclude. $\square$