Pseudo double categories of manifolds, cobordisms and diffeomorphisms of cobordisms

We construct, for each non-negative integer n, a monoidal pseudo double
category of manifolds (closed, smooth and of dimension n), cobordisms, and
isotopy classes of certain (collar-compatible) isotopy classes diffeomorphisms
between cobordisms. As a first step towards a combinatorialisation of the
construction we construct a (strict) double category of nested tuples and
permutations, which we believe is equivalent to a full subcategory of our
geometric pseudo double category, when n = 2.