Natural Transformations and Parametricity In Theorems for Free!, Wadler says that the characterization of parametricity can be re-expressed in terms of lax natural transformations and this will be the subject of a further paper. Which paper is he referring to?

The categorical approach to paramtericty I know uses dinatural transformations as in Functorial Polymorphism by Bainbridge, Freyd, Scedrov and P.J. Scott. What is the connection between lax natural transformation and dinatural transformation formulations of parametricity?
Unfortunately, the remark of Wadler is too cryptic for me to tell what use he wanted to make of "lax natural transformations".  Here is a guess.  Relation-preservation squares can often be recast as lax commutative squares.  This is how they used to be written in old automata theory papers/books.  See paragraph 1.2 in my Notes on Semigroups.  To do this kind of thing, you have to mix up relations and morphisms and pretend that they are the same.  I am also not sure that it buys you anything new.  It is just uglier notation for saying the same thing as relation-preservation.

Please feel free to explore the connection, but I am not confident that you will find anything new by doing it.