Let $X$, $Y$, and $Z$ be smooth manifolds, and let $C_1 \subset T^*X \setminus 0 \times T^*Y \setminus 0$ and $C_2 \subset T^*Y \setminus 0 \times T^*Z \setminus 0$ be conic canonical relations, with the usual twisted cotangent convention for operator kernels. Let
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\begin{align*}
A \in I^{m_1}(X,Y;C_1)
\end{align*}
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and
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\begin{align*}
B \in I^{m_2}(Y,Z;C_2)
\end{align*}
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be properly supported Fourier integral operators acting on half-densities, in the half-density operator-order convention used in this chapter.
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Assume that $C_1$ and $C_2$ compose transversally over $T^*Y \setminus 0$, that the projection from the fibre product
is an embedded conic canonical relation on that microlocal region. Then the composition $AB$ is a properly supported Fourier integral operator satisfying
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\begin{align*}
AB \in I^{m_1+m_2}(X,Z;C_1 \circ C_2).
\end{align*}
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Moreover, after fixing the standard Maslov and half-density identifications on the transversal fibre product, the principal symbol of $AB$ is obtained from the exterior product of the principal symbols of $A$ and $B$ by multiplication followed by half-density contraction along the fibre over $T^*Y$.