Lemma 88.23.11. In the situation above. If $X$ is locally Noetherian, $f$ is separated and locally of finite type, and $U' \to U$ is a monomorphism, then $\Delta _{f_{/T}}$ is rig-surjective.
Proof. The diagonal $\Delta _ f : X' \to X' \times _ X X'$ is a closed immersion and the restriction $U' \to U' \times _ U U'$ of $\Delta _ f$ is surjective. Hence the lemma follows from the discussion in Remark 88.23.6 and Lemma 88.23.10. $\square$
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