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Mathematically, monads are governed by set of laws that should hold for the monadic operations. This idea of laws is not unique to monads: Haskell includes other operations that are governed, at least informally, by laws. For example, x /= y and not (x == y) ought to be the same for any type of values being compared. However, there is no guarantee of this: both == and /= are separate methods in the Eq class and there is no way to assure that == and =/ are related in this manner. In the same sense, the monadic laws presented here are not enforced by Haskell, but ought be obeyed by any instances of a monadic class. The monad laws give insight into the underlying structure of monads: by examining these laws, we hope to give a feel for how monads are used.

The Functor class, already discussed in section 5, defines a single operation: fmap. The map function applies an operation to the objects inside a container (polymorphic types can be thought of as containers for values of another type), returning a container of the same shape. These laws apply to fmap in the class Functor:

fmap id = id
fmap (f . g) = fmap f . fmap g

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