I am currently reading about train track maps (specifically the following text: Bestvina & Handel: Train tracks and automorphisms of free groups). In this text, they consider a map $f:G \to G$ where $G$ is a graph and $f$ is a so called topological representative of an outer automorphism of some free group $F_n$. A connected graph has a universal cover $\Gamma$, together with a projection $p: \Gamma \to G$. The text then mentions a lift of this map $f:G \to G$ as being a map $\tilde{f}: \Gamma \to \Gamma$ (no conditions on this map $\tilde{f}$ are defined in the text). However, I am only familiar with the following definition of lift:
a lift of a map $f:G \to G$ is a map $\tilde{f}: G \to \Gamma$ such that $p \circ \tilde{f} = f$.
Can anyone explain what the definition from the text should be or give me a link to some explanation on this 'different' notion of lift?
EDIT the only 'guess' I can make, is that the definition used in the text means that the diagram $$\require{AMScd} \begin{CD} \Gamma @>{\tilde{f}}>> \Gamma\\ @V{p}VV @V{p}VV \\ G @>{f}>> G \end{CD}$$ is commutative...
The meaning is exactly what you guessed. More generally, if $f:X\to Y$ and you have chosen maps $X'\to X$ and $Y'\to Y$, a lift of $f$ to a map $X'\to Y'$ is a map which makes the square commute. Even more generally, the term "lift" can refer to any sort of operation where you "raise" something backwards along maps (from the codomain up to the domain). The intended meaning is usually clear from context.