Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 1 / 20I / a / Solution

Use path concatenation in the order of traversal and let $\bar{u}(t)=u(1-t)$. A loop $\alpha$ based at $x_0$ determines the loop $\bar{u}\ast \alpha \ast u$ based at $x_1$. Thus the change of basepoint isomorphism isA based homotopy of $\alpha$ can be concatenated with the fixed outer paths to give a based homotopy of the resulting loops, so this map is well-defined on the fundamental group. Different parenthesizations and affine reparametrizations of path concatenation are based homotopic.