Definition 7

The cylinder of the mapping $f$ is called and is denoted by Cyl($f$), the topological space quotient $U/R$.

Denote by $\alpha_f: X\times \mathbf{I}\rightarrow$ Cyl($f$) and $\beta_f: Y\rightarrow$ Cyl($f$) the restrictions to $U_1$ and $U_2$ of the canonical surjection of U onto $U/R$. The mapping $\alpha_f$ is a homotopy and its term is $\beta_f\circ f$.

Proposition 4 (Universal Property of Cylinders)

Let Z be a topological space, let $\beta : Y\rightarrow Z$ be a continuous mapping and let $\alpha : X\times \mathbf{I}\rightarrow Z$ be a homotopy whose term is $\beta \circ f$. There exists a unique continuous mapping $\varphi$ from Cyl($f$) into Z such that $\alpha =\varphi \circ \alpha_f$ and $\beta =\varphi \circ \beta_f$.

The mapping $\psi$ from U into Z which coincides with $\alpha$ in $X\times \mathbf{I}$ and with $\beta$ in Y is continuous. We have $\alpha (x,1) =\beta (f(x))$ for every $x\in X$, hence $\psi$ defines by passing to the quotient a continuous mapping $\varphi$ from Cyl($f$) into Z such that $\alpha =\varphi \circ \alpha_f$ and $\beta =\varphi \circ \beta_f$. Since the images of $\alpha_f$ and $\beta_f$ cover Cyl($f$)$,\varphi$ is the only mapping from Cyl($f$) into Z satisfying these relations.