$d'$ so that $B:= d A^{-1}$ and $B':= d' (A')^{-1}$ are matrices with integer entries. Then we can form a “tree” diagram of Shioda maps: $$\xymatrix{
    &   &       &   X_{dd'I} \ar@{-->}[dll]_{\phi_{d'I}} \ar@{-->}[drr]^{\phi_{dI}} &   &   & \\   &X_{dI} \ar@{-->}[dl]_{\phi_B} \ar@{-->}[dr]^{\phi_{B^T}}&&&&X_{d'I}\ar@{-->}[dl]_{\phi_{B'}} \ar@{-->}[dr]^{\phi_{(B')^T}}&\\
X_A &   & X_{A^T} &                                                   & X_{A'}& &X_{(A')^T}
        }$$

One can then calculate that $\phi_M \circ \phi_{cI} = \phi_{cM}$ for any integer valued matrix $M$ and positive integer $c$. This means we can simplify our tree to just the diagram: $$\xymatrix{
    &   &       &   X_{dd'I} \ar@{-->}[dlll]_{\phi_{d'B}} \ar@{-->}[dl]^{\phi_{d'B^T}} \ar@{-->}[dr]_{\phi_{dB'}} \ar@{-->}[drrr]^{\phi_{d(B')^T}} &    &   & \\   
X_A &   & X_{A^T} &                                                   & X_{A'}& &X_{(A')^T}
        }$$

The Calabi-Yau orbifolds are just finite quotients of the hypersurfaces $X_A$, $X_{A^T}$, $X_{A'}$ and $X_{(A')^T}$, so we can view them in the context of the diagram: $$\xymatrix{
    &   &       &   X_{dd'I} \ar@{-->}[dlll]_{\phi_{d'B}} \ar@{-->}[dl]^{\phi_{d'B^T}} \ar@{-->}[dr]_{\phi_{dB'}} \ar@{-->}[drrr]^{\phi_{d(B')^T}} &    &   & \\   
X_A \ar[d]& & X_{A^T}\ar[d] &                                                 & X_{A'}\ar[d]&   &X_{(A')^T}\ar[d] \\
Z_{A,G} && Z_{A^T, G^T} && Z_{A', G'} && Z_{(A')^T, (G')^T}
        }$$ Letting $H$ and $H'$ be the groups $H:= (\phi_{d'B})_*^{-1}(G)$ and $H':= (\phi_{dB'})_*^{-1}(G')$, we know that:

\[biratprop\] The following birational equivalences hold: $$\begin{aligned}
Z_{A,G} &\simeq X_{dd'I}/(H/J_{F_{dd'I}}); \\
Z_{A^T, G^T} & \simeq X_{dd'I}/ (H^{\perp_{d'B}}/J_{F_{dd'I}}); \\
Z_{A', G'} &\simeq X_{dd'I}/(H'/J_{F_{dd'I}}); \text{ and 