Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 137 / 3 / b / Solution

A weight-zero modular function is invariant under $\Gamma$, so it descends uniquely through the quotient map to a meromorphic function on $\Gamma \backslash \mathfrak{h}$. Its assumed meromorphic Fourier expansion at every cusp makes the descended function meromorphic in each cusp coordinate. A meromorphic function on a compact Riemann surface is equivalently a holomorphic morphism to the Riemann sphere, sending each pole to infinity. Thus there is a morphismwith $f={\varphi }_f{|}_{\Gamma \backslash \mathfrak{h}}\circ \pi$. The open quotient $\Gamma \backslash \mathfrak{h}$ is dense in $X(\Gamma )$, so this identity also proves uniqueness.