 Solution

True. A path $u:x\to y$ gives the basepoint-change isomorphism of fundamental groups $[\gamma ]\mapsto [\overline{u}\ast \gamma \ast u]$. Concatenating paths is compatible with homotopy and loop multiplication. Using the reversed path gives its inverse, because $u\ast \overline{u}$ and $\overline{u}\ast u$ contract relative to endpoints.