Let $f:C_0\to \mathbb{C}$ be a holomorphic map, and suppose that there exists a disk chain along a continuous map $\gamma :[t_0,t_1]$. Then we define $\tilde{f}:C^{\prime }\to \mathbb{C}$ to be the analytic continuation of $f$ by any disk chain which runs along $\gamma$. By the last lemma, this is well defined.