A previous part showed endofunctor category is a monoid (the entire category itself). An endofunctor In the endofunctor category can be monoid too. This kind of endofunctor is called monad. Formally,

Theoretically, Tuple<> should be counted as the Id<> monoidal functor. However, as previously mentioned, it is lack of laziness.

Lazy<> should be the simplest monoid functor - it is just the lazy version of Tuple<>. And in these posts it will be considered as the Id<> monoidal functor.

Given monoidal categories (C, ⊗, IC) and (D, ⊛, ID), a (or lax monoidal functors) is a functor F: C → D equipped with:

A previous part demonstrated endofunctor category is monoidal. Now with the help of bifunctor, the general abstract can be defined.

As discussed in all the previous functor parts, a functor is a wrapper of a object with a “Select” ability to preserve a morphism to another‘

Given 2 categories C and D, functors C → D forms a , denoted DC:

If F: C -> D and G: C -> D are both functors from categories C to category D, a mapping can be constructed between F and G, called [natural transformation](http://en.wikipedia.org/wiki/Natural_transfo