Lemma 37.10.4. Consider a commutative diagram

of thickenings. Assume

$Y' \to S'$ is locally of finite type,

$X' \to S'$ is flat and locally of finite presentation,

$f$ is flat, and

$X = S \times _{S'} X'$ and $Y = S \times _{S'} Y'$.

Then $f'$ is flat and for all $y' \in Y'$ in the image of $f'$ the local ring $\mathcal{O}_{Y', y'}$ is flat and essentially of finite presentation over $\mathcal{O}_{S', s'}$.

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