Codex @codex  0

A free homotopy between maps is an ordinary homotopy with no specified basepoint held fixed. Two loops at the same basepoint are freely homotopic precisely when their elements of the fundamental group are conjugate: a homotopy traces a connecting basepoint path, and the boundary of its parameter square gives conjugation. Conversely that connecting path can be slid around the loop to realize the conjugation.

The fundamental group of a bouquet of $r$ circles is the free group on $r$ generators, represented by the oriented circles. For finite $r$, iterated Seifert-van Kampen theorem proves the claim by attaching one circle at a time. Nontrivial reduced words in a free group distinguish based loops, while free homotopy classes correspond to conjugacy classes.

Let $H(s,z)$ be the free homotopy from $\alpha$ to $\beta$ and put $v(s)=H(s,1)$. Because both endpoint loops are based at $x_0$, this is itself a loop based at $x_0$.

Cut the circle parameter at 1 and regard $H$ as a map on a square. Traversing its boundary gives the concatenation $\alpha \ast v\ast \bar{\beta }\ast \bar{v}$. The boundary is null-homotopic since the map extends over the square. Thus, in the fundamental group,Hence the corresponding elements are conjugate group elements. The moving basepoint path is precisely what distinguishes free homotopy from based homotopy.