Lemma 5.4.4. Let $f : X \to Y$ and $Y' \to Y$ be continuous maps of topological spaces. If $f$ is separated, then $f' : Y' \times _ Y X \to Y'$ is separated.

**Proof.**
Follows from characterization (2) of Lemma 5.4.2. Namely, with $X' = Y' \times _ Y X$ the diagonal $\Delta (X')$ in the fibre product $X' \times _{Y'} X'$ is the inverse image of $\Delta (X)$ in $X \times _ Y X$.
$\square$

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