Trivial loop on the $1$-Skeleton

Question

Following Hatcher's proof of Hurewicz Theorem (version of 1999) we arrive at the point that we must show that the loop in the picture, created following the path $0,1,2,3,1,0,1,3,0,3,2,0,2,1,0$, is triavial on the $1$-skeleton. He says that it is "obviously" trivial, but unfortunately my $3$D-visualization is probably not enough good to catch the point. Is there a proper way to understand such obvious statement?

Thanks!

Answer 1

Doesn't "trivial" here just mean that we can collapse "$-a-b-a-$" to "$-a-$" repeatedly and eventually end up with a loop of length $0$?

0,1,2,3,1,0,1,3,0,3,2,0,2,1,0
         \ /
  0,1,2,3,1,3,0,3,2,0,2,1,0
         \ /
    0,1,2,3,0,3,2,0,2,1,0
           \ /
      0,1,2,3,2,0,2,1,0
           \ /
        0,1,2,0,2,1,0
             \ /
          0,1,2,1,0
             \ /
            0,1,0
             \ /
              0

Answer 2

Do you see how a path such as $a,b,a$ is trivial? I.e. If you go from $a$ to $b$ and then back to $a$, we might as well just stay at $a$. Using that, we can collapse the $1,0,1$ to $1$, the $3,0,3$ to $3$, and the $2,0,2$ to $2$. We are left with $0,1,2,3,1,3,2,1,0$. That can then be reduced to $0,1,2,3,2,1,0$, then to $0,1,2,1,0$, then to $0,1,0$, then to $0$.