Back in the days of HYLO, it was common to write hylomorphisms with an additional natural transformation in them. Well, I was still coding in evil imperative languages back then, but I have it on reliable, er.. well supposition, that this is probably the case, or at least that they liked to do it back in the HYLO papers anyways.
Transcoding the category theory mumbo-jumbo into Haskell, so I can have a larger audience, we get the following 'frat combinator' -- you can blame Jules Bean from #haskell for that.
hyloEta :: Functor f =>
(g b -> b) ->
(forall a. f a -> g a) ->
(a -> f a)
hyloEta phi eta psi = phi . eta . fmap (hyloEta phi eta psi) . psi
We placed eta in the middle of the argument list because it is evocative of the fact that it occurs between phi and psi, and because that seems to be where everyone else puts it.
Now, clearly, we could roll eta into phi and get the more traditional hylo where f = g. Less obviously we could roll it into psi because it is a natural transformation and so the following diagram commutes:
This 'Hylo Shift' property (mentioned in that same paper) allows us to move the 'eta' term into the phi term or into the psi term as we see fit. Since we can move the eta term around and it adds no value to the combinator, it quietly returned to the void from whence it came. hyloEta offers us no more power than hylo, so out it goes.
So, if its dead, why talk about it?