Functor

A functor $F:\mathcal{C}\to \mathcal{D}$ assigns:

• Objects: $A\mapsto F(A)$
• Morphisms: $f:A\to B\mapsto F(f):F(A)\to F(B)$

Preserves:

• Identity: $F({\text{id}}_A)={\text{id}}_{F(A)}$
• Composition: $F(g\circ f)=F(g)\circ F(f)$

See also

• Category
• Natural Transformation
• Adjunction
• Monad