A contravariant functor from $\mathcal{C}$ to $\mathcal{D}$ is a functor from $\mathcal{C}^{\mathsf{op}}$ to $\mathcal{D}$.
11.5.2 Contravariant Functors
Let $\mathcal{C}$ and $\mathcal{D}$ be categories, and let $\mathcal{C}^{\mathsf{op}}$ denote the opposite category of $\mathcal{C}$ of 
, .
In detail, a contravariant functor from $\mathcal{C}$ to $\mathcal{D}$ consists of:
-
1.
Action on Objects. A map of sets
\[ F \colon \operatorname {\mathrm{Obj}}(\mathcal{C}) \to \operatorname {\mathrm{Obj}}(\mathcal{D}), \]called the action on objects of $F$.
-
2.
Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, a map
\[ F_{A,B} \colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B) \to \operatorname {\mathrm{Hom}}_{\mathcal{D}}(F(B),F(A)), \]called the action on morphisms of $F$ at $(A,B)$.
satisfying the following conditions:
-
1.
Preservation of Identities. For each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the diagram
commutes, i.e. we have
\[ F(\operatorname {\mathrm{id}}_{A}) = \operatorname {\mathrm{id}}_{F(A)}. \] -
2.
Preservation of Composition. For each $A,B,C\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the diagram
commutes, i.e. for each composable pair $(g,f)$ of morphisms of $\mathcal{C}$, we have
\[ F(g\circ f) = F(f)\circ F(g). \]
Throughout this work we will not use the term “contravariant” functor, speaking instead simply of functors $F\colon \mathcal{C}^{\mathsf{op}}\to \mathcal{D}$. We will usually, however, write
\[ F_{A,B}\colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,B)\to \operatorname {\mathrm{Hom}}_{\mathcal{D}}(F(B),F(A)) \]
for the action on morphisms
\[ F_{A,B}\colon \operatorname {\mathrm{Hom}}_{\mathcal{C}^{\mathsf{op}}}(A,B)\to \operatorname {\mathrm{Hom}}_{\mathcal{D}}(F(A),F(B)) \]
of $F$, as well as write $F(g\circ f)=F(f)\circ F(g)$.
