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

Tuple<> looks like the simplest functor by just wrapping a value. It is most close to the [Identity functor of Haskell](http://hackage.haskell.org/package/transformers-0.4.3.0/docs/Data-Functor-Identi

A simple functor in DotNet category is Lazy<>. Its Select functions can be easily implemented:

A F: C → D is a structure-preserving ) from category C to category D:

An individual monoid (T, ⊙, I) can be a category M:

A monoid, denoted a 3-tuple (M, ⊙, I), is a set M with

This post and the following posts will introduce category theory and its important concepts via C# and LINQ, including functor, applicative functor, monoid, monad, etc. Categories were first introduce

p is the ) of function F :