maps are birational to quotient maps as well. Note that Bini requires $d$ to be the smallest positive integer so that $dA^{-1}$ is an integral matrix, but the requirement that $d$ is the smallest such integer is unnecessary. One can just use the first part of Theorem 2.6 of [@BvGK] to eliminate this hypothesis.

An upshot of this reinterpretation of the theorem is that the mirror statement of BHK duality is a relation of two orbifolds birational to different orbifold quotients of the same Fermat hypersurface in projective space. Namely, $X_A/ \tilde G$ is birational to $X_{dI}/ ( ((\phi_B)_*)^{-1}(J_{F_A})+ H/J_{F_{dI}})$ while $X_{A^T}/\tilde G^T$ is birational to $X_{dI}/ (((\phi_{B^T})_*)^{-1}(J_{F_{A^T}})+ H^{\perp_B}/J_{F_{dI}})$. As $J_{F_A} \subseteq G$ and $J_{F_{A^T}} \subseteq G^T$,

$$\begin{aligned}
((\phi_B)_*)^{-1}(J_{F_A}) \subseteq ((\phi_B)_*)^{-1}(G) = H; \text{ and } \\((\phi_{B^T})_*)^{-1}(J_{F_{A^T}}) \subseteq ((\phi_{B^T})_*)^{-1}(G^T) \subseteq H^{\perp_B}
\end{aligned}$$

which gives us the following corollary:

\[binicor\] The Calabi-Yau orbifold $Z_{A,G}$ is birational to $X_{dI}/ ( H/J_{F_{dI}})$ and its BHK mirror $Z_{A^T,G^T}$ is birational to $X_{dI}/ (H^{\perp_B}/J_{F_{dI}})$.

Multiple Mirrors
================

As stated in the introduction, one can take two polynomials $$F_A = \sum_{i=0}^n \prod_{j=0}^n x_j^{a_{ij}}; \qquad F_{A'} = \sum_{i=0}^n \prod_{j=0}^n x_j^{a_{ij}'}.$$ that cut out two hypersurfaces in weighted-projective $n$-spaces, $X_A \subseteq W{\mathbb{P}}^n(q_0,\ldots, q_n)$ and $X_{A^T} \subseteq W{\mathbb{P}}^n(q_0',\ldots, q_0')$, 