In the paper "Simple curves on hyperbolic tori" by McShane and Rivin, they show that if $T$ is a hyperbolic once punctured torus, one can define a norm on the homology $H_1(T,\mathbb{Z})$ by associating, to every primitive class (nonzero, and not a multiple of another class), the length of the unique simple closed geodesic in that homology class.

In Theorem 3.2, they study the unit ball $\mathcal{B}_\ell(T)$ as we vary the hyperbolic structure on $T$. In the proof, they say the following: "Pick a simple geodesic $\gamma$, and let $\gamma'$ be the shortest associated generator of the fundamental group".

In Corollary 1 (right below the theorem), they seem to imply that $\gamma'$ is also a geodesic. They call it the "conjugate geodesic". I don't understand what they mean by $\gamma'$. Isn't there a unique geodesic in every free homotopy class of curves (equivalently, in every conjugacy class of the fundamental group)?