What is a lift? What exactly is a lift? I wanted to prove that for appropriately chosen topological groups $G$ we can show that the completion of $\widehat{G}$ is isomorphic to the inverse limit $$\lim_{\longleftarrow} G/G_n$$
I wasn't sure how to construct the map so I asked this question to which I got an answer but I don't understand it. Searching for "lift" or "inverse limit lift" has not helped and I was quite sure that if $[x] \in \widehat{G}$ I could just "project" $x_n$ to $$\lim_{\longleftarrow} G/G_n$$ using the map $\prod_{n=1}^\infty \pi_n (x) = \prod_{n=1}^\infty x_n$ in $G/G_n$. Would someone help me finish this construction and explain to me what a lift is? Thank you.
Edit
I don't know any category theory. An explanation understandable by someone not knowing any category theory would be very kind. 
 A: The term lift is merely meant to mean the following. Given a surjective map $G\to G'$ a lift of an element $x\in G'$ is a choice of $y\in G$ such that $y\mapsto x$ under this map.
In the linked answer you have $(x_n)\in \lim G/G_n$. Thus the notation has $x_n\in G/G_n$. The elements $y_n$ are just chosen preimages of $x_n$ under the natural surjection $G\to G/G_n$.
A: See the Wikipedia article on lift:
In a category $C$, let be given two arrows $f,g$ with codomain $Z$. A lift (of $f$ with respect to $g$) is an arrow $F$ such that $g\circ F =f$. So it is a morphism from $f$ to $g$ in the over category $C/Z$.
Motivation: the idea is that $g:X\to Z$ indicates the vertical  direction pointing down: $X$ lies above $Z$ via $g$. Think of $g$ as a fibre bundle, or covering map, or vertical projction. So a given $f:Y\to Z$ takes values in the lower space; and to lift $f$ (via $g$) means to 'let it take values in the upper space': find a 'modified' $F:Y\to X$ such that, after vertically projecting its values, we get $f$ back; in other words $p\circ F=f$. If we take $Y=[0,1]$ this is path-lifting. 
In particular:


*

*a lift of the identity $id:Z\to Z$ (via $g:X\to Z$) is an arrow $s:Z\to X$ such that $g\circ s=id$, i.e. a section of $g$.


(I believe in your case this is meant: a section of a quotient map $q:X\to X/\sim$ is a map $s:X/\sim\to X$ such that $s[x]\sim x$. One could say that $s[x]\in X$ is a lift of $[x]$.)
DUALLY: let be given two arrows $f,g$ with domain $Z$. A lift of $f$ (via $g$) is an arrow $F$ such that $F\circ f=g$. So it is a morphism from $f$ to $g$ in the under category $Z/C$.
Motivation: again the idea is that $g:Z\to X$ indicates the vertical  direction, but now pointing up: $X$ lies above $Z$ via $g$. Think of $g$ as an 
inclusion; then:


*

*a lift of $f:Y\to X$ (via an inclusion $g:Z\to X$) is an arrow $F:Y\to X$ such that $F\circ i=g$, i.e. an extension of $f$.


Related concepts are injective and projective objects. Moreover note that lifts need not exist in general (see also Zhen Lin's comment).
