The Stacks project

Remark 88.23.6. In the situation above consider the diagonal morphisms $\Delta _ f : X' \to X' \times _ X X'$ and $\Delta _{f_{/T}} : X'_{/T'} \to X'_{/T'} \times _{X_{/T}} X'_{/T'}$. It is easy to see that

\[ X'_{/T'} \times _{X_{/T}} X'_{/T'} = (X' \times _ X X')_{/T''} \]

as subfunctors of $X' \times _ X X'$ where $T'' \subset |X' \times _ X X'|$ is the inverse image of $T$. Hence we see that $\Delta _{f_{/T}} = (\Delta _ f)_{/T''}$. We will use this below to show that properties of $\Delta _ f$ are inherited by $\Delta _{f_{/T}}$.

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