Functor

Let $C$ and $D$ be categorys. A functor $F$ from $C$ to $D$ is a mapping that

• associates each object $X\in ob(C)$ an object $F(X)\in ob(D)$.
• associates each morphism f : X → Y in $C$ to a morphism F ( f ) : F ( X ) → F ( Y ) in $D$ such that the following two conditions hold:
  -- $F(i{d}_X)=i{d}_{F(X)}$ for every $X\in ob(C)$.
  -- $F(g\circ f)=F(g)\circ F(f)$ for all morphisms $f:X\to Y$ and $g:Y\to Z$ in $C$.
  That is, functors must preserve identity morphisms and composition of morphisms.

As a result, this defines a category of categories and functors – the objects are categories, and the morphisms (between categories) are functors.

An important relationship between functors is adjunction.