Suppose $D$ be a quaternion over the number field $K$.
$\Bbb{Definition.}$ The Eichler ordre $R$ of level $N$ in $D$ is defined as the one which satisfies the following equality with the completion $K_v$ of $K$ at all places $v$$\colon$ \begin{equation*} R \otimes_{O_K} \!{O_{K_v}} \cong \Bigg\{ \begin{pmatrix}\label{matrix} a & b \\ c & d \\ \end{pmatrix} \in {\mathrm M}_2(O_{K_v}) ~|~ c ~\equiv~ 0~ {\mathrm{mod}} ~ N \Bigg\}. \end{equation*}
Let us write $\widehat{R} = R \otimes_{\Bbb Z} \widehat{\Bbb Z}$, where $\widehat{\Bbb Z} = \underset{n > 1}{\varprojlim} \, {\Bbb Z}/n{\Bbb Z}$. In the similar manner, we define $\widehat{D} \colon= D \otimes_{\Bbb Z} \widehat{\Bbb Z}$.
Let us once and for all choose one Eichler ordre $R$ of level $N$ and denote by $N(\widehat{R}^{\times})$ the normaliser of $\widehat{R}^{\times}$ in $(\widehat{D})^{\times}$. We say to Eichler ordres $R$ and $R'$ are conjugate to each other if there is some element $a \in D$ such that $R' = aRa^{-1}$.
Q. Why conjugate classes of all Eichler ordres of level $N$ in $D$ is classified by $N(\widehat{R}^{\times})\setminus(\widehat{D})^{\times}/D^{\times}$?