Let $X$ and $Y$ be smooth manifolds, let $\Omega_X^{1/2}$ and $\Omega_Y^{1/2}$ denote their half-density bundles, let $m \in \mathbb{R}$, and let $\Gamma_Y \subset T^*Y \setminus 0$ and $\Gamma_X \subset T^*X \setminus 0$ be conic open sets. Let
be a properly supported microlocal Fourier integral operator $A \in I^m(X,Y;C)$ acting on half-densities near $(p_0,q_0)$. Suppose that $A$ is elliptic at $(p_0,q_0) \in C$, meaning that its principal symbol $\sigma_m(A)(p_0,q_0)$ is nonzero and hence invertible in the Maslov and half-density symbol line $L_C$ over $C$ under the standard half-density principal-symbol pairing for graph canonical relations.
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Use the convention that $\Psi^0(M)$ denotes the properly supported order-$0$ pseudodifferential operators on half-densities over a smooth manifold $M$, and that an operator is smoothing microlocally on a conic set $V \subset T^*M \setminus 0$ when, after applying pseudodifferential cutoffs elliptic on $V$, its Schwartz kernel is smooth in the corresponding microlocal region.
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Then, after inserting microlocal cutoffs supported in sufficiently small conic neighbourhoods of $q_0$ and $p_0$ so that the relevant compositions are properly supported in the microlocal region under consideration, there exist conic neighbourhoods $V_Y \subset \Gamma_Y$ of $q_0$ and $V_X \subset \Gamma_X$ of $p_0$, with $\kappa(V_Y) = V_X$, and a Fourier integral operator