Proposition 2 (Lifting of homotopies)

Let B be a topological space and let E be a covering of B. Let $B'$ be a topological space and let $f_0$ and $f_1$ be continuous mappings of $B'$ into B. Let $\sigma : B'\times$ $\mathbf{I}\rightarrow B$ be a homotopy joining $f_0$ to $f_1$. Let $g_0: B'\rightarrow E$ be a continuous lifting of $f_0$ to E. There exists a unique continuous mapping $\widetilde{\sigma}: B'\times \mathbf{I}\rightarrow$ E lifting $\sigma$ which is a homotopy with initial term $g_0$. Its term $g_1: B'\rightarrow E$ is a continuous lifting of $f_1$ to E.

This follows from corollary 3 of prop. 8 of I, p. 130, applied to the simply connected and locally connected topological space $\mathbf{I}$.