Definition 8.1 (Covariant functor):

Given two categories ${\cal C}_1$ and ${\cal C}_2$, a covariant functor 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$(h(A), h(B))$ such that the following 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$(B, C)$,

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

(ii)

$h($id$_{A}) =$   id$_{h(A)}$