Explicit form of a lift $\tilde f: \tilde X_1 \to \tilde X_2$ of a continuous map $f: X_1 \to X_2$

Question

This is embarrassingly simple for most, but I am a High School student trying to teach myself, and I am having trouble figuring it out:

In the post Basic question about lifting maps to covering spaces it is claimed that any continuous map $f: X_1 \to X_2$ "lifts" to a map $\tilde f: \tilde X_1 \to \tilde X_2$ (provided that $X_1$ and $X_2$ have universal covers).

I tried, but I am having trouble finding an explicit form for the map $ \tilde f$ by trying to cook up commutative diagrams and using lifting properties.

I looked for related answers and it seems we can also use deck transformations and conjugation properties of the group of deck transformations, but I still feel stuck here.

Answer 1

We have a projection $p_1: \tilde X_1 \to X_1$, so first write $g = fp_1: \tilde X_1 \to X_2$. Now $\tilde X_1$ is simply connected, so this lifts to the universal cover of $X_2$; call the lift $\tilde f: \tilde X_1 \to \tilde X_2$. So we have a commutative diagram

$$\require{AMScd}\begin{CD} \tilde X_1 @>\tilde f >> \tilde X_2\\ @Vp_1 VV @Vp_2 VV\\ X_1 @>f>> X_2 \end{CD}$$

as desired.