Let $X$ and $Y$ be smooth manifolds, let $\pi_X:X\times Y\to X$ and $\pi_Y:X\times Y\to Y$ be the coordinate projections, and let $K\in \mathcal D'(X\times Y)$ be the distribution kernel of a Fourier integral operator from $Y$ to $X$. Define the primed kernel wave front relation by
\begin{align*}
C\circ \operatorname{WF}(f):=\{(x,\xi)\in T^*X\setminus 0:\text{ there exists }(y,\eta)\in \operatorname{WF}(f)\text{ with }(x,\xi;y,\eta)\in C\}
\end{align*}
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\begin{align*}
\operatorname{WF}'_X(K):=\{(x,\xi)\in T^*X\setminus 0:\text{ there exists }y\in Y\text{ such that }(x,\xi;y,0)\in \operatorname{WF}'(K)\}.
\end{align*}