 understand each other about the category $\mathfrak{A}$.  If $\mathfrak{A_O^{\alpha}}$=$\mathfrak{A_O^{\beta}}$ and $A_O^{\alpha}=A_O^{\beta}$ and $A_I^{\alpha}=A_I^{\beta}$ and $\mathfrak{A_I^{\alpha}}$=$\mathfrak{A_I^{\beta}}$