Let $π_1(X; x_0 → x_1)$ be the set of homotopy classes of paths from $x_0$ to $x_1$ (preserving end points). Prove that there is a bijective correspondence between $π_1(X,x_0)$ and $π_1(X;x_0 → x_1 )$
Probably it's implicit in the problem that $x_0$ and $x_1$ are in the same path -component of X. Otherwise it's clearly not true in general.
My attempt:
Tried to consider a path $\gamma : [0,1] \to X$ by, $\gamma(0)=x_0$ and $\gamma(1)=x_1$ and then $\bar{\gamma} : [0,1] \to X$ by, $\bar{\gamma}(t) = \gamma(1-t)$ and then concatenate $\gamma * \bar{\gamma}$ to obtain a loop in X based at $x_0$.
But unable to figure out one-one correspondence. How to get the other way around?
While posting the question I found this but it precisely states my attempt as hint.
Thanks in Advance for help!
Fix $[\gamma]\in \pi_1(X,x_0\rightarrow x_1)$. Consider the maps $$f:\pi_1(X,x_0)\ni[\alpha]\mapsto[\alpha]\ast[\gamma]\in\pi_1(X,x_0\rightarrow x_1) $$ and
$$g:\pi_1(X,x_0\rightarrow x_1)\ni[\beta]\mapsto[\beta]\ast[\overline\gamma]\in\pi_1(X,x_0) .$$
Since $f\circ g$ and $g\circ f$ are the identity maps, $f$ and $g$ are bijections.