A quick/geometric reason why Hatcher's reparameterizations work in the proof of Proposition 1.3? During the proof of Proposition 1.3 in Hatcher, Algebraic Topology, (the result that $\pi_1(X,x_0)$ is a group with respect to the product $[f][g]=[f\cdot g]$) he uses some reparamaterization tricks I'm not very comfortable with. Take associativity - he asserts that $$f\cdot(g\cdot h)=\Big[(f\cdot g)\cdot h\Big]\circ\phi$$ where $\phi$ is a map created by a graph appearing on page 27. Of course, it is probably possible to just write out $\phi$ explicitly and verify the composite, but why intuitively does this work?
 A: Why do so many topology texts insist that paths in $X$ have to be maps $[0,1] \to X$? Alternatives are used in Crowell and Fox  "Knot theory", and in my book, Topology and Groupoids, first edition, differently titled, 1968. 
You can define a path in $X$ of length $r \geqslant 0$ to be either a map $[0,r] \to X$ or a pair  $(r,f)$, called a "Moore path",  where $f:[0,\infty) \to X$ is constant on $[r,\infty)$. In both cases, composition of paths is associative, and in fact yields a category. You then need to discuss reparametrisation, and homotopies. Students find it easy to think that a path (journey) of length $r$ followed by a path of length $s$ is of length $r+s$. 
June 14, 2018: In view of other answers, I should also point out that it is useful to discuss the category, say  $P(X)$, of paths on a space  and to consider  pushouts of such categories in the case $X$ is a union of open sets. as a step to the van Kampen Theorem. 
A: I don't have the book here, but intuitively the product $f\cdot g$ in $\pi_1$ is just the concatenation of the loops $f$ and $g$. How exactly you parametrize it is not really important, since everything is only up to homotopy. So you can think of $f \cdot g \cdot h$ as the loop which first traverses $f$, then $g$, then $h$, making associativity immediately clear.
A: A lot of the pictures in algebraic topology only really make sense after you already understand the subject. To prove that the composition law on $\pi_1$ is associative, I think it might be more beneficial for the intuition to prove that both loops $(f\cdot g)\cdot h$ and $f\cdot (g\cdot h)$ are homotopic to the loop
$$
s\mapsto \begin{cases} f(3s) & \text{if $0\leq s \leq 1/3$} \\ g(3s - 1) & \text{if $1/3 \leq s \leq 2/3$} \\ h(3s - 2) & \text{if $2/3\leq s \leq 1$}. \end{cases}
$$
