Let $q_X=(x,\xi)\in T^*X\setminus 0$ and $q_Y=(y,\eta)\in T^*Y\setminus 0$. Suppose there are conic open neighbourhoods $\Gamma_X\subset T^*X\setminus 0$ of $q_X$ and $\Gamma_Y\subset T^*Y\setminus 0$ of $q_Y$ such that:
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1. There is a homogeneous canonical diffeomorphism
Equivalently, for every $s\in\mathbb{R}$, under the order convention in which an order-$m$ Fourier integral operator maps microlocal $H^s$ regularity on the source branch to microlocal $H^{s-m}$ regularity on the target branch,