Lemma 32.4.9. In Situation 32.4.5. Suppose we are given an $i$ and a morphism $T \to S_ i$ such that

$T \times _{S_ i} S = \emptyset $, and

$T$ is quasi-compact.

Then $T \times _{S_ i} S_{i'} = \emptyset $ for all sufficiently large $i'$.

Lemma 32.4.9. In Situation 32.4.5. Suppose we are given an $i$ and a morphism $T \to S_ i$ such that

$T \times _{S_ i} S = \emptyset $, and

$T$ is quasi-compact.

Then $T \times _{S_ i} S_{i'} = \emptyset $ for all sufficiently large $i'$.

**Proof.**
By Lemma 32.2.3 we see that $T \times _{S_ i} S = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} T \times _{S_ i} S_{i'}$. Hence the result follows from Lemma 32.4.3.
$\square$

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