I've seen in many places that one can define the usual Hecke operator $T_p$ via correspondences by $(\pi_1)_*(\pi_2^*(\bullet))$ in the diagram $$Y \xleftarrow{\pi_1} Y_0(p) \xrightarrow{\pi_2} Y$$ where $Y$ has trivial level structure at $p$, $Y_0(p)$ is the modular curve given by adding $\Gamma_0(p)$ level to $Y$, $\pi_1$ is the map which forgets the $p$-subgroup on the level of moduli, and $\pi_2$ is the map quotienting by the $p$-subgroup on the level of moduli.
I'd like to see a reference for the analogous correspondence for $U_p$, when $Y$ has nontrivial level structure at $p$ already, especially in the case where $Y$ has full $p$-level structure. I have a guess as to what it is, but a reference would be very helpful, especially if it also described the compatibility between $U_p$ and $T_p$ geometrically. Can anyone give such a reference?