$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\char{char}$I am reading a note on Teichmüller space, and I come across a somewhat algebraic problem in the picture below, which may be easy to experts.

Teichmüller space $T_g$ is naturally a manifold homeomorphic to $\mathbb R^{6g - 6}$, and $\Gamma_g$ acts properly discontinuously on $T_g$. Thus, $M_g$ is an orbifold with $\pi_1^\text{orb}(M_g) = \Gamma_g$.

We are able to see the topology in two ways:

By Representation theory:

$$T_g \hookrightarrow \Hom(\pi_1(S), \PSL_2(\mathbb R))/\PSL_2(\mathbb R) = \char_2(\pi_1(S)),$$

where the image of $T_g$ is the open subset of discrete and faithful representations. A simple counting argument shows

$$\dim(\Gamma_g) = \dim \char_2(G) = (2g - 1)*3 - 3 = 6g - 6.$$

By Fenchel–Nielson Coordinates:

The definition of the Teichmüller space $T_g$ I used here is the following (the "isotopic" can be replaced by "homotopic" and the "homeomorphism" can be replaced by "orientation-preserving diffeomorphism" since any homeomorphism is homotopic to an diffeomorphsim between surfaces and the resulting space is the same ):

Suppose S is a topological surface of genus $g\geq 2$. A marked Riemann surface $(X, f)$ is a Riemann surface X together with a homemorphism $f:S\rightarrow X$. Two marked surfaces $ (X,f)\sim (Y,g)$ are equivalent if $gf^{-1}: X\rightarrow Y$ is isotopic to an isomorphism(biholomorphism). The Teichmüller space $T_g$ is given by:

$$T_g=\{(X,f)\}/\sim$$

I wonder how to understand this injective map:

$$T_{g}\hookrightarrow \Hom(\pi_{1}({S}),\PSL_{2}(\mathbb{R}))/\PSL_{2}(\mathbb{R})$$

explicitly?

(All I can understand is just an injective map from the fundamental group to $\operatorname{Aut}(\mathbb{H})$ just like what the author did here: Teichmuller space as Discrete Faithful Representations up to Conjugation.)

I'm also wonder how to compute $\Hom(\pi_1(S), \PSL_2(\mathbb R))/\PSL_2(\mathbb R)$, why it's $\char_{2}(\pi_{1}(S))$?

And how can I compute $\char_{2}(\pi_{1}(S))$?

I find something that looks just like the right hand of the mapping, which is called the representation of surface group, but I can't find a formal explanation or proof about how can it be related with Teichmüller space. Can anybody help me? Any advice or comment is welcome.