Let $S$ be a compact smooth surface.
On pg 276 of Farb and Margalit's A Primer on Mapping Class Groups, the following definition of the Teichmuller space of $S$ is given.
A hyperbolic sturcture on $S$ is a diffeomorphism $\phi:S\to X$, where $X$ is a surface with a finite area hyperbolic metric. The pair $(X, \phi)$ is called a marking. Two markings $(X, \phi_1:S\to X_1)$ and $(S, \phi_2:S\to X_2)$ are said to be homotopic if there is an isometry $I:X_1\to X_2$ such that $I\circ \phi_1$ is homotopic to $\phi_2$.
Definition 1. The Teichmuller space of $S$ is the set of homotopy classes of all the markings on $S$.
On pg. 277, the authors give an alternate definition, as follows.
Let $H(S)$ be the set of all the hyperbolic metrics on $S$, and let the diffeomorphism group $\text{Diff}_0(S)$ of $S$ act on $H(S)$ by pullback.
Definition 2. The Teichmuller space of $S$ is defined as the orbit space of $H(S)$ under the action of $\text{Diff}_0(S)$.
I am unable to see how the two definitions are equivalent. A Natural thing to do is the following.
Let $M(S)$ denote the set of all the markings of $S$. Define a map $M(S)\to H(S)$ by sending $(S, \phi:S\to X)$ to $\phi^*(g_X)$, where $g_X$ denotes the metric on $X$. Then I need to show that if $(S, \phi_1)$ and $(S, \phi_2)$ represent the same element in $H(S)/\text{Diff}_0(S)$, then $(S, \phi_1)$ and $(S, \phi_2)$ are homotopic. But I am unable to see why this should be true.
Hint: What's missing in the logic is an important theorem of surface topology:
Sometimes the second conclusion is expressed by saying that $h_1,h_2$ are "diffeotopic", although that is a rare term. On the other hand, the topological version of this theorem is almost always stated using the analogous terminology "isotopic": two self homeomorphisms of a surface are homotopic if and only if they are isotopic.