The Stacks project

Here's the proof: The result follows from Categories, Lemma 4.17.2 applied to the observation that the functor $s^*:Y/S\to X/S$, sending an object $Y\to Y'$ to an object $X\xrightarrow{s}Y\to Y'$, is cofinal. To show that $s^*$ is cofinal, we check the conditions of #9830. Suppose $X\to X'$ is an object of $X/S$. Then there are morphisms $Y\to Y'\leftarrow X'$ such that the square commutes and $X\to Y''$ is in $S$. Since $S$ is saturated, also $Y\to Y'$ lies in $S$. On the other hand, suppose we have a commutative diagram $$\begin{matrix} X&\xrightarrow{s}&Y\\ \downarrow&&\downarrow\\ X'&\rightrightarrows&Y' \end{matrix}$$ Since $X/S$ is filtered, there is a morphism $Y'\to Y''$ coequalizing $X'\rightrightarrows Y'$ and such that the composite $X\to X'\rightrightarrows Y'\to Y''$ lies in $S$. Since $S$ is saturated, $Y'\to Y''$ lies also in $S$, thus $Y\to Y'\to Y''$ is an object in $Y/S$ and we win.