Waldhausen's algebraic K-theory of manifolds satisfies a homotopical lift of the classical h-cobordism theorem and provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. I will give an overview of joint work with Goodwillie, Igusa and Malkiewich about a homotopical lift of an equivariant h-cobordism theorem. In this talk I will zoom in on the functoriality of the stable equivariant h-cobordism space and I will give a flavor of some of the geometric ideas that go into these constructions.