I want to understand the adelic point of view on automorphic forms. I use Gelbart's notations: $\mathbb{A}$ for the adeles, $G$ for the group $GL_2$, $K_\infty$ for $SO(2)$, $K_0^N$ for the adelic analogue of the congruence subgroup $\Gamma(N)$. By strong approximation, $$G_{\mathbb{A}} = G_{\mathbb{Q}} G_\infty^+ K_0^N$$
and that $$G_\mathbb{Q} \cap G_\infty^+ \prod_p K_p^N = \Gamma_0(N)$$
How does this imply $$Z_\mathbb{A}G_{\mathbb{Q}} \backslash G_\mathbb{A} / K_0^N K_\infty \simeq \Gamma_0(N) \backslash SL_2(\mathbb{R}) / SO(2)?$$