Let $S_g$ denote the closed orientable surface with genus $g\geq 2$. A marking of $S_g$ is a diffeomorphism $\phi:S_g\to X$, where $X$ carries a hyperbolic metric on it. Two markings $\phi:S_g\to X$ and $\psi:S_g\to Y$ are said to be homotopic if there is an isometry $I:X\to Y$ such that $I\circ \phi$ is homotopic to $\psi$. The Teichmuller space of $S_g$, written $\text{Teich}(S_g)$, is defined as the set of all the markings of $S_g$ up to homotopy.
Let $\mathcal D$ denote the set of all the discrete faithful representations $\pi_1(S_g)\to \text{PSL}_2(\mathbf R)$. The group $\text{PGL}_2(\mathbf R)$ acts on $\mathcal D$ by conjugation.
I want to show that there is a bijection $\text{Teich}(S_g)\leftrightarrow \mathcal D/\text{PGL}_2(\mathbf R)$.
Let $\text{Mark}(S_g)$ denote the set of all the markings of $S_g$. Define a map $\alpha:\text{Mark}(S_g)\to \mathcal D/\text{PGL}_2(\mathbf R)$ as follows: For $(X, \phi)\in \text{Mark}(S_g)$, we have $\alpha((X, \phi))$ is the conjugacy class of the composite
$$\pi_1(S_g)\xrightarrow{\phi_*}\pi_1(X)\xrightarrow{\cong}\text{Deck}(\tilde X\to X)\xrightarrow{\eta_*}\text{Deck}(\mathbf H^2\to X)\hookrightarrow \text{Isom}^+(\mathbf H)= \text{PSL}_2(\mathbf R)$$
Here $\tilde X\to X$ is a metric universal cover of $X$ and $\eta:\tilde X\to \mathbf H^2$ is an isometry.
Question. I want to see why $\alpha$ factors through $\text{Teich}(S_g)$.
At the very least, the following should be true: If $(X, \phi)$ is a marking of $S_g$, and $I:X\to X$ is an isometry, then $\alpha((X, \phi))=\alpha((X, I\circ \phi))$, which is to say that the conjugacy class of the composite
$$\pi_1(S_g)\xrightarrow{\phi_*}\pi_1(X)\xrightarrow{I_*} \pi_1(X)\xrightarrow{\cong}\text{Deck}(\tilde X\to X)\xrightarrow{\eta_*}\text{Deck}(\mathbf H^2\to X)\hookrightarrow \text{Isom}^+(\mathbf H)$$
is same as the conjugacy class of the composite above. But I cannot see how to show this.