User:Викидим/A monad is just a monoid in the category of endofunctors, what's the problem?

Airline route map

An oversimplified example of a category is an airline route map: the objects of the category are cities (like Los Angeles, Las Vegas, Denver, etc. on the illustration) and the morphisms are one-way flights (like Los Angeles → Las Vegas). The composition is taking two flights: for example, a composition of Los Angeles → Las Vegas and Las Vegas → Denver is a morphism Los Angeles → Denver. The identity morphism (staying put) naturally is available for every city-object.

The second fundamental concept of category theory is the concept of a functor, which plays the role of a morphism between two categories C₁ and C₂: it maps objects of C₁ to objects of C₂ and morphisms of C₁ to morphisms of C₂ in such a way that sources are mapped to sources, and targets are mapped to targets (or, in the case of a contravariant functor, sources are mapped to targets and vice-versa). A special (and most often used in applications) type of functor maps the category to itself ("endofunctor").