Remark 2

Suppose that the mapping $f$ is proper. Let P be a point of Cyl($f$). If P = $\alpha_f(x, t)$, with 0 $\leqslant t <$ 1 and $x\in$ X, we have $\alpha^{-1}_f(P) = (x, t)$. In the contrary case, there exists $y\in Y$ such that $P =\beta_f(y)$ and $\alpha^{-1}_f(P) =\overset{-1}{f}(y)\times \{1\}$. This proves that the fibres of the mapping $\alpha_f$ are quasi-compact. The mapping $\alpha_f$ is then proper (TG, I, p. 75, th. 1), since it is closed.

By prop. 5 and TG, I, p. 72, prop. 2, the mapping $\beta_f$ is itself proper. It follows that, if $f$ is proper, the canonical surjection of U onto Cyl($f$) is proper and hence, in particular, universally strict (I, p. 20, corollary).