Serre-type eigenvalue correspondence for definite quaternion algebras

Let $D$ and $\bar D$ be the quaternion algebras introduced earlier, let $\Sigma$ be the specified finite set of places, let $\mathbf{F}$ be the space of mod-$p$ algebraic functions on $\bar D^\times(F)\setminus\bar D^\times(\mathbb{A}^f)$, and let $\mathbf{T}^{D}(K_\Sigma)$ and $\mathbf{T}^{\bar D}_\Sigma$ be the corresponding mod-$p$ Hecke algebras, with Hecke operators $T_w$ for $w\notin\Sigma$. Serre-type eigenvalue correspondence. The systems of eigenvalues for $(T_w)$ of $\mathbf{T}^{\bar D}_\Sigma$ acting on $\mathbf{F}$ are in bijection with the systems of eigenvalues for $(T_w)$ of $\mathbf{T}^{D}(K_\Sigma)$ arising from mod-$p$ modular forms. This is presented as a generalization of Serre's theorem and as a comparison between mod-$p$ Hecke eigenvalues for the two quaternion algebras; the source gives no resolution.