Let $\mathbb{H} = \{x+iy: y > 0\}$ be usual the complex upper half plane.
The modular group is defined to be $$PSL(2,\mathbb{R}) = SL(2,\mathbb{R})/\{I_2, -I_2\}.$$ So it is a class of matrices where $M$ and $-M$ are identified.
Similarly, a modular surface is $$PSL(2,\mathbb{Z}) \backslash \mathbb{H}$$ which I believe the same thing as $$\mathbb{H} / SL(2,\mathbb{Z}).$$ Not totally sure how moding occur since $\mathbb{H} \subseteq \mathbb{C}$ and $SL(2,\mathbb{Z}) \subseteq SL(2,\mathbb{R}),$ they are not the same object "a complex points" and a "matrix".
Thebn there is a fact, which most text/documant about hyperbolic geometry mentioned without proof
"we can identify the unit tangent bundle on modular surface with $$SL(2,\mathbb{R}) / SL(2,\mathbb{Z})"$$
(for example one good text available online is https://webusers.imj-prg.fr/~antonin.guilloux/notes.pdf?fbclid=IwAR0SN5LhHC_yOWn0_Hx6fQ_NDK6aPYHFi07rdH3ycp8quDH6wxV-7330dzQ and it mentioned this fact briefly on page 7)
So identify here mean a bijection between two sets ? I do not quite understand how to see this. Can anyone give some help please ?