Definition 8.2 (Contravariant functor):

A contravariant functor between the two given categories ${\cal C}_1$ and ${\cal C}_2$ is a rule that assigns to each object $A \in {\cal C}_1$ an object $h(A) \in {\cal C}_2$ and to each morphism $f \in$   Mor$(A, B)$, where $A, B$ are objects in ${\cal C}_1$, a unique morphism $h(f) \in$   Mor$(h(B), h(A))$ such that the following conditions hold:

(i)

Given objects $A, B$ and $C$ in ${\cal C}_1$ and a pair of morphisms $f \in$   Mor$(A, B)$, $g \in$Mor$(B, C)$,

$$h(g\circ f) = h(f)\circ h(g)$$

(ii)

$h($id$_{A}) =$   id$_{h(A)}$