= 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 $\bar u*\alpha*u$ based at $x_1$. Thus the <change of basepoint isomorphism> is
$$
\boxed{u_\#([\alpha])=[\bar u*\alpha*u]}.
$$
A <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.
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