**Proof.**
The question is local on $X$, hence we may assume $X$, $Y$, and $S$ are affine. Arguing as in the proof of More on Algebra, Lemma 15.81.13 we can find a commutative diagram

\[ \xymatrix{ X \ar[r]_ i \ar[d]_ f & \mathbf{A}^ m_ Y \ar[r]_ j \ar[ld]^ p & \mathbf{A}^{n + m}_ S \ar[ld] \\ Y \ar[r] & \mathbf{A}^ n_ S } \]

The assumption that $\mathcal{O}_ Y$ is pseudo-coherent relative to $S$ implies that $\mathcal{O}_{\mathbf{A}^ m_ Y}$ is pseudo-coherent relative to $\mathbf{A}^ m_ S$ (by flat base change; this can be seen by using for example Lemma 37.56.15). This in turn implies that $j_*\mathcal{O}_{\mathbf{A}^ n_ Y}$ is pseudo-coherent as an $\mathcal{O}_{\mathbf{A}^{n + m}_ S}$-module. Then the equivalence of the lemma follows from Derived Categories of Schemes, Lemma 36.12.5.
$\square$

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