contravariant functor

Let \(\mathcal{C}, \mathcal{D}\) be categories. A functor \(F: \mathcal{C} \rightarrow \mathcal{D}\) is a map

• sending an object \(C \in \mathrm{Ob}(\mathcal{C})\) to an object \(F(C) \in \mathrm{Ob}(\mathcal{D})\)
• sending a morphism \(f \in \mathrm{Hom}_{\mathcal{C}}(A,B)\) to a morphism \(F(f) \in \mathrm{Hom}_{\mathcal{D}}(F(B),F(A))\)

such that \(F\) preserves the identity morphism and composition of morphisms:

\begin{align*} F(\mathrm{id}_{A}) =& \mathrm{id}_{F(A)} \\ F(f \circ g) =& F(g) \circ F(g) \end{align*}