Example 1

Let $X',Y'$ be topological spaces and let $f': X'\rightarrow Y'$ be a continuous mapping. Let $u: X\rightarrow X'$ and $v: Y\rightarrow Y'$ be continuous mappings such that $f'\circ u=v\circ f$. There exists a unique continuous mapping $w:$ Cyl($f$)$\rightarrow$ Cyl($f'$) such that $w\circ \alpha_f=\alpha_{f'}\circ (u\times$ Id$_{\mathbf{I}})$ and $w\circ \beta_f=\beta_{f'}\circ v$.

According to prop. 4, applied to Z = Y and $\beta =$ Id$_Y$, there exists a unique continuous mapping $\gamma_f:$ Cyl($f$)$\rightarrow Y$ such that $\gamma_f(\alpha_f(x, s)) =f(x)$ and $\gamma_f(\beta_f(y)) =y$ for $x\in X,s\in \mathbf{I}$ and $y\in Y$.