Functor

A functor is a Morphism between Categories

If you have Category $A$ and Category $B$, a functor

• Maps every Object in $A$ to an object in $B$
  • $O\in A⟹F(O)\in B$
• Maps every Morphism in $A$ to a morphism in $B$
  • $f:O_1\to O_2⟹F(f):F(O_1)\to F(O_2)$
• Preserves Structure
  • Preserving Identity: $F({\text{id}}_O)={\text{id}}_{F(O)}$
  • Preserving Composition: $F(g\circ f)=F(g)\circ F(f)$