= Change of basepoint isomorphism
{title2=$u_\#$}
Let $u$ be a path from $x_0$ to $x_1$. Traversing paths from left to right gives an <isomorphism>
$$
u_\#: \pi_1(X,x_0)\longrightarrow\pi_1(X,x_1),\qquad[\alpha]\longmapsto[\bar u*\alpha*u].
$$
Concatenation and cancellation of a path with its reverse prove the homomorphism property and show that $\bar u$ supplies its inverse. A different connecting path changes this isomorphism by an inner automorphism, so it need not be canonical.
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