Lemma 86.20.1. In the situation above. If $f$ is locally of finite type, then $f_{/T}$ is locally of finite type.

**Proof.**
(Finite type morphisms of formal algebraic spaces are discussed in Formal Spaces, Section 85.20.) 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 65.23.3 we conclude.
$\square$

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