I was reading a survey on hyperbolic surfaces by Javier Aramayona. One theorem states that any complete connected hyperbolic surfaces up to isometry corresponds to torsion-free Fuchsian groups up to conjugation. This is because the hyperbolic structure of a surface $X$ induced the isometry in the developing map $D:\tilde{X}\to\mathbb{H}$ by the choice of chart $U_0$. Different choices will induce conjugate holonomy representations Hol: $\pi_1(X,p_0)\to PSL_2(\mathbb{R})$ and Hol: $\pi_1(X,p_1)\to PSL_2(\mathbb{R})$ as the fundamental groups with different base points are conjugate.
However, different developing maps differ by an element of $PSL_2(\mathbb{R})$ i.e. $D=g\circ D'$. I am confused on why it becomes conjugation when we consider input as loops i.e. holonomy representation.