So considering a Monad to be a Triple (T:C -> C, η, µ) with eta and mu as the Natural transformations with appropriate signatures, isn't this in essence an F-Algebra?
My thinking is that being both (Monad and F-Alg) centered around some particular action(s) on|of a given Endofunctor, what stops Monads NOT just being special cases of F-Algebra's? (Also for some context I've been reading a bit of Homotopy theory & Bart Jacobs so I like could just be you know not seeing clearly and making connections where there are none...) I'm seeing a Monad as an F-Algebra of some Monoidal category or Quasicategory with the action of the Endofunctor or alternatively the operations definable on the Algebra of the Endofunctor as the particular semantics of that F-Algebra (e.g. what makes a Monad the State Monad or the Identity Monad or what have you instead...)
Examples, Proofs etc. in Scala or Haskell are good, but I might need clarification if it's in Haskell.