Distributional kernels and typed contractions

Write ${{\mathcal{E}}^{\prime }}_{\Gamma }(A)$ for compact distributions supported in $A$ with wavefront in the closed cone $\Gamma$, with the normal microlocal topology. It is not identified with a smooth Fréchet stage. Pullbacks and proper pushforwards are continuous on their stated stages; general distribution products have the appropriate hypocontinuity properties BDH.