Question about the first proof in Hatcher's Algebraic Topology

This steps seems to be unnecessary. In what follows, I'll assume you have p. 30 in front of you (since it's available freely online).

Start with the part where he has constructed neighborhood $N$ of $y_0 \in Y$ and a partition $0 = t_0 < \dots t_n = 1$ such that $F(N \times [t_i,t_{i+1}]) \subset U_i$ for each $i = 0, \dots, n-1$. The lift is constructed by defining $\widetilde{F}:N \times I \to \mathbb{R}$ to be $h_{i} \circ F$, where $h_{i}$ is the homeomorphism from the evenly covered neighborhood $U_i$ to the sheet $\widetilde{U_i}$ which contains $h_{i-1} \circ F(y_0,t_i)$. (The homeomorphism $h_0$ is determined by the given lift $\widetilde{F}$ at time $t=0$.)

The only question is whether this is well-defined. One might worry that $N \times \{t_i\}$ is mapped by the $(i-1)$-st lift into the sheet $\widetilde{U}_{i-1}$ but that the neighborhood $N$ needs to be modified so that the $i$-th lift glues to this. In other words, could it be that we need to replace $N$ with a smaller neighborhood $N'$ so that $h_{(i-1)} \circ F(N' \times \{t_i\}) \subset \widetilde{U}_{i-1} \cap \widetilde{U}_{i}$?

This is not the case because, by design, $F(N \times \{t_i\}) \subset U_{i-1} \cap U_i$. Therefore, the map above is well-defined. The lift is continuous by the glueing lemma for open sets.