The path functor is defined here:
Now, it doesn't mention it, but it seems clear that Path functor would map continuous mapping $f: X \rightarrow Y$ to post composition by $f$.
Now, the problem asks
Natural automorphisms of the path functor $Path: Top \rightarrow Set$ would be given by
$$\alpha: Path \rightarrow Path$$ such that for topological space $X$, If $g$ is a path in $X$, and $f: X \rightarrow Y$ is a continuous map,
$$f\alpha_X(g) = \alpha_Y(fg)$$
The book says that this is called re-parametrization, but I'm not sure how this coincides with the intuitive definition. I'm not even sure how the naturality condition would say that $g$ and $\alpha_X(g)$ even have the same endpoints.
And to tackle the main question, would it be the right approach to consider the forgestful functor from Top to Set?
Thanks!
By Yoneda, each natural transformation $\alpha:\textbf{Path}\to\textbf{Path}$ is induced by a map $\phi:[0,1]\to[0,1]$ in $\textbf{Top}$, that is a continuous map. Then $\alpha$ is a natural isomorphism iff $\phi$ is a homeomorphism.
So take $\phi:[0,1]\to[0,1]$ to be a homeomorphism. The corresponding $\alpha$ takes a path $p:[0,1]\to X$ to the path $p_1=p\circ\phi$, that is $$p_1(t)=p(\phi(t)).$$