If C is a category, a monad on C consists of a functor together with two natural transformations: (where 1C denotes the identity functor on C) and (where T2 is the functor from C to C). These are required to fulfill the following conditions (sometimes called coherence conditions):
- (as natural transformations );
- (as natural transformations ; here 1T denotes the identity transformation from T to T).
We can rewrite these conditions using following commutative diagrams:
See the article on natural transformations for the explanation of the notations Tμ and μT, or see below the commutative diagrams not using these notions:
The first axiom is akin to the associativity in monoids, the second axiom to the existence of an identity element. Indeed, a monad on C can alternatively be defined as a monoid in the category whose objects are the endofunctors of C and whose morphisms are the natural transformations between them, with the monoidal structure induced by the composition of endofunctors.
