Theorems for Free: The Monad Edition

This is for the record, since the derivations took me a while and I'd rather not lose them. A functor is the signature: module type FUNCTOR = sig type 'a t val fmap : ('a -> 'b) -> ('a t -> 'b t) end satisfying the following laws: Identity: fmap id ≡ id Composition: fmap (f ∘ g) ≡ fmap f ∘ fmap g An applicative structure or idiom is the signature: module type APPLICATIVE = sig type 'a t val pure :