Theorem 7.2 (Reparametrization theorem):

Let $X$ be a topological space. Suppose that $\phi : [0, 1] \longrightarrow [0, 1]$ is a continuous map such that $\phi(0) = 0$ and $\phi(1) = 1$. Then for any given path $\gamma$ in $X$, we have a homotopy

$$\gamma \sim \gamma\circ\phi$$