Let $\mathbb{H}$ be hyperbolic plane, $\Gamma$ is a discrete subgroup of $PSL_2(\mathbb{R}$) so that $\Gamma \backslash \mathbb{H}$ is a compact hyperbolic surface. Maybe it will be very simple to you but I am very confused when I try to construct a homeomorphism $\phi: \Gamma \backslash T_1 \mathbb{H} \longrightarrow T_1(\Gamma \backslash \mathbb{H})$

Suppose $\Gamma (z,v)$ is an element of $\Gamma \backslash T_1 \mathbb{H}$, so what is $\phi (\Gamma (z,v))$? Actually I read in some book and they always accept this fact obviously, but for me it's not obvious.

I'm so sorry if it is not a question of researching.