  We construct an $(\infty,1)$-functor that takes each smooth $G$-manifold with
corners $M$ to the space of equivariant smooth $h$-cobordisms ${\mathcal
H}_{\mathrm{Diff}}(M)$. We also give a stable analogue ${\mathcal H}^{\mathcal
U}_{\mathrm{Diff}}(M)$ where the manifolds are stabilized with respect to
representation discs. The functor structure is subtle to construct, and relies
on several new ideas. In the non-equivariant case $G=e$, our
$(\infty,1)$-functor agrees with previous constructions of the smooth
$h$-cobordism space as a functor to the homotopy category.
