I'm reading Rohrlich's article about modular curves in the Cornell-Silverman-Stevens' book Modular forms and Fermat's Last Theorem. I read the proof of the following theorem about description of the Hecke correspondence on the modular curve as a moduli space of elliptic curves: 
The author defines modular curve by defining its function field first (as a subfield of certain extension of $\mathbb{Q}(t)$) and use curve-function field correspondence. Along the proof, here are the lines that I can't fully understand.
- which follows from the compatibility of reduction at a good place with base extension.
In the article, Hecke correspondence on $X_{0}(N)$ is defined as a triple $(X_{0}(N, p), \varphi_{p}, \psi_{p})$ where $\varphi_{p}: X_{0}(N, p) \to X_{0}(N)$ simply corresponds to the inclusion of the function fields (the function field for $X_{0}(N, p)$ is a certain extension of it of $X_{0}(N)$ which is fixed by a smaller subgroup). At a glance, it seems that the reduced curve $E_{z} / \mathbb{C}$ corresponds to $z \in X_{0}(N)$ is a reduction $E_{\varphi_{p}(z)}/\mathbb{C}$ of some base extension of the curve $E/\mathcal{O}_{z}$, but I'm not sure what the author actually intended.
- by the compatibility of reduction with isogenies.
I actually don't get why the reduction and isogeny are compatible and why it implies the equation (2).