Functor

Definition

Suppose ${\mathcal {C}},{\mathcal {D}}$ are categories. A functor (also called covariant functor) ${\mathcal {F}}$ from ${\mathcal {C}}$ to ${\mathcal {D}}$ comprises the following data:

• A mapping ${\mathcal {F}}:\operatorname {Ob} {\mathcal {C}}\to \operatorname {Ob} {\mathcal {D}}$.
• For any $A,B\in {\mathcal {C}}$, a mapping ${\mathcal {F}}:{\mathcal {C}}(A,B)\to {\mathcal {D}}(FA,FB)$.

satisfying the following condition:

• For any $A\in {\mathcal {C}}$, ${\mathcal {F}}(\operatorname {id} _{A})=\operatorname {id} _{F(A)}$.
• For any $A,B,C\in {\mathcal {C}}$, and $f\in {\mathcal {C}}(A,B),g\in {\mathcal {C}}(B,C)$, we have $F(g\circ f)=Fg\circ Ff$.

There is a related notion of contravariant functor. Note that a contravariant functor between two categories is not a functor between the categories. To emphasize that a functor is a functor and not a contravariant functor, we use the term covariant functor.