I am reading "The Tits alternative for $\operatorname{Out}(F_n)$ I: Dynamics of exponentially-growing automorphisms" by Bestvina, Feighn and Handel. In this paper, the following is written:
Suppose that $G$ is marked graph and that $K$ is a subgraph whose non-contractible components are $C_1, \ldots, C_l$. Choose vertices $v_i \in C_i$ and a maximal tree $T \subset G$ such that each $T \cap C_i$ is a maximal tree in $C_i$. The tree $T$ determines inclusions $\pi_1(C_i, v_i) \to \pi_1(G,v)$. Let $F^i \subset F_n$ be the free factor of $F_n$ determined by $\pi_1(C_i, v_i)$ under the identification of $\pi_1(G,v)$ with $F_n$. Then $F^1 \ast \ldots \ast F^l$ is a free factor of $F_n$. Without a specific choice of $T$, the $C_i$ only determine the $F^i$ up to conjugacy.
(bold text is my emphasis). I tried 'proving' this last sentence, but only manage to make some intuitive (partial) reasoning: suppose the edges of $G$ are labeled, then the edges which are not in the tree $T$ become the generators of $F_n$. If we then consider a component $C_i$, the edges not in $T \cap C_i$ generate a free group. If we take another such tree, we obtain other generators for $F_n$ and for $C_i$. I tried proving that the generators of $C_i$ in the second case are conjugated to the ones in the first case, but could not conclude. Any help would be appreciated.
The inclusion $\iota_{v_i,T}:\pi_1(C_i,v_i)\to\pi_1(G,v)$ can be defined by sending the equivalence class $[\gamma]$ of a loop $\gamma$ based at $v_i$ to $[p_{v_i,T}^{-1}\gamma p_{v_i,T}],$ where $p_{v_i,T}$ is the unique path from $v$ to $v_i$ in $T,$ juxtaposition means concatenating the paths, and the $^{-1}$ means the reverse path.
For another choice $T',v_i'$ we get a different inclusion $\iota_{v_i',T'}:\pi_1(C_i,v_i')\to\pi_1(G,v).$ For any path $\eta$ from $v_i'$ to $v_i$ in $C_i,$ there is an isomorphism $\eta^*:\pi_1(C_i,v_i)\to\pi_1(C_i,v_i')$ sending $[\gamma]$ to $[\eta^{-1}\gamma\eta].$ The images of the different inclusions are conjugate (in a non-canonical way depending on $\eta$): send $[\gamma]$ in the image of $\iota_{v_i,T}$ to $[p_{v_i',T'}^{-1}\eta^{-1}p_{v_i,T}\gamma p_{v_i,T}^{-1} \eta p_{v_i',T'}]$ in the image of $\iota_{v_i',T'}.$