Reference request: Hecke operators acting as correspondences

I'm trying to see that the Hecke algebra defined as $\mathbb{Q}[\text{GL}_2(\mathbb{Z}_p)\backslash \text{GL}_2(\mathbb{Q}_p)/\text{GL}_2(\mathbb{Z}_p)]$ maps to the ring of correspondences $\text{Corr}_\sim^0(M_n,M_n)$ where $M_n$ is the modular curve of elliptic curves with full $n$-torsion structure. I defined this space over $\mathbb{Z}[1/n]$, but for the above result I'm happy to work with $(M_n\otimes\mathbb{C})^{an}$. I didn't find any reference that didn't just put the definition for the generators (essentially assuming that it is a ring homomorphism). My idea was to define, for $g\in\text{GL}_2(\mathbb{Q}_p)$, the correspondence $\Gamma_{\pi\circ g}*\Gamma_{\pi}^t$, where $\pi$ is the projection from the modular curve $M_{n,p}$ of elliptic curves with $n$-torsion structure and a $p$-cyclic subgroup (at least in the interesting cases). I have problems to verify that $T_p^2$ follows the usual relation $T_{p^2}+pR_p*T_p$. Is there any book or paper that does this explicitly?