They don’t need it since their only arrows are the identities. Since they only need concern themselves with identities, an equality GADT serves them in better stead.

This is also why their definition of an HFunctor drops ffmap. It eases their derivation. For instance in http://comonad.com/haskell/category-extras/src/Control/Functor/HigherOrder/Composition.hs

I have to deal with a lot of newtype noise to implement ffmap, since they don’t need it at all, they don’t have to cram that into the paper.

If you look at Johann and Ghani’s earlier paper “Initial Semantics is Enough!” at http://crab.rutgers.edu/~pjohann/tlca07-rev.pdf you’ll find the exact definition above (modulo the choice of newtyping a tuple vs. using a data type.

As for the derivations I was planning to do a post on that at some point, but the short answer is they are based on the definition of Kan extensions as (co)ends, which can be found readily in the Wikipedia article on Kan extensions or more formally in Categories for the Working Mathematician.

One question: can you explain how you converted the definition of the Kan extension and its universal property into the Haskell datatype above?

It should be unambiguous, correct? I’m confused as to why Johann and Ghani’s GADT paper in POPL 2008 here gives a different encoding of the left Kan extension.

Any help would be appreciated.