It could well be that Vaughan and I are defining the midpoint structure in the same way. Here's how I described it (using the conventions from my last posting). Let F:I --> I v I be a final coalgebra. We will denote the top of I as T and the bottom as B. Construct the "halving" map, h:I --> I, (on [-1,1] it will send x to x/2) as: T v F v B F'v F' F' I --> 1 v I v 1 ------> I v I v I v I ---> I v I --> I where F' denotes the inverse of F, and, by a little overloading, T and B denote the maps constantly equal to T and B. The leftmost map records the fact that the terminator is a unit for the ordered-wedge functor. Let g be the endo-function on I x I defined recursively by: g<x,y> = if dx = T and dy = T then <x,y> else if dx = T and uy = B then h(g(dx,uy>) else if ux = B and dy = T then h(g<ux,dy>) else if ux = B and uy = B then <x,y>. The values of g lie in the first and third quadrants, that is, those points such that either dx = dy = T or ux = uy = B. The two maps g d x d g u x u I x I --> I x I --> I x I and I x I --> I x I --> I x I give a coalgebra structure on I x I. The midpoint operation may be defined as the induced map to the final coalgebra.