Lifting correspondence in Algebraic topology Recently I have been studying algebraic topology and came across the notion of lifting correspondence. Here, lifting correspondence definition is the same that Munkres uses in his book. However, I cannot get any intuition behind that notion. I am well aware of the definitions about "lifts", and covering spaces (but maybe I don't actually know what that "means" too). I have seen a question regarding intuitions behind lifts but that didn't help me because it mainly focused on what it is used for. I want to know what it actually "means." Can anyone please provide the intuition (if there is any) behind those notions? Or is that something that needs to be worked from the definitions without indulging into its meaning?
Definition: Let $p: E\rightarrow B$ be a covering map; let $b_0\in B$. Choose $e_0$ so that $p(e_0)= b_0$. Given an element $[f]$ of $\pi_1(B,b_0)$, Let $\tilde{f}$ be the lifting of $f$ to a path in $E$ that begins at $e_0$. Let $\phi([f])$ denote the endpoint $\tilde{f}(1)$ of $\tilde{f}$. then $\phi$ is well defined set map: $\phi: \pi_1(B,b_0)\rightarrow p^{-1}(b_0)$. Here $\phi$ is said to be the lifting correspondence derived from covering map $p$
 A: First we will start with what we mean when we talk about a lift.
Consider a map $p:E\to B$ (not necessarily a covering map) . $f$ is a continuous mapping of some space $X$ into $B$. A lifting of $f$ is a map $f': X \to E$ such that $p\circ f=f'$.
Intuitively this means that if we have $f:X\to B$ take values in the space $B$ there exists this special map $f'$ which will take this map $f$ to the above space $E$, such that when it is composed $p$ it gives me back $f$. For concrete examples you can refer Hatcher.
Lifting Correspondence: I owe this part of the answer a lot to anon.
Let me first explicitly write out the definition:
Let $p:E\to B$ be a covering map(I guess you know what I mean by a covering map); let $b_0\in B$. Chose $e_0$ such that $p(e_0)=b_0$. Given an element $[f]\in \pi_{1}(B,b_0)$ ,let $f'$ denote the lifting of $f$ to a path in $E$ which begins at $e_0$. Let $g([f])$ denote the end point of $f'(1)$ of $f'$. Then $g$ is a well defined map $$g:\pi_{1}(B,b_0)\to p^{-1}(b_0)$$This is what we mean by the lifting correspondence.
Intuition: Till now we were considering paths which were lifted to certain paths in the above  covering space . What happens when we consider a loop? It might get me into some new place in the above covering space. So for this map $g$ to be well defined we have to consider the lift of its original position (that is, its base point), which just happens to be the preimage of that point under the covering map.
Now comes the question why do we happen to require it? We require it because you might have seen that when we find the fundamental group of $S^1$ we use the lifting correspondence and show that it is bijective and a homomorphism.
