Definition 3

Let C and $C'$ be categories. A functor from C into $C'$ is a morphism of quivers $\varphi : C\rightarrow C'$ such that $\varphi (f g) =$ $\varphi (f)\varphi (g)$ for every pair $(f, g)$ of composable maps of C.

Let C and $C'$ be categories and let $\varphi : C\rightarrow C'$ be a functor. Let $f$ be an arrow of C$,a$ its origin and $b$ its end; then, $\varphi (f)$ is an arrow of $C'$ with origin $\varphi (a)$ and end $\varphi (b)$.

Let C, $C',C''$ be categories and let $\varphi : C\rightarrow C'$ and $\varphi ': C'\rightarrow C''$ be functors. Then, $\varphi '\circ \varphi$ is a functor.

Let C be a category. Then the quiver morphism Id$_C$ is a functor.

Let $\varphi : C\rightarrow C'$ be a functor. In order that $\varphi$ be an isomorphism, it is necessary and sufficient that the mappings Som($\varphi$ ) and Fl($\varphi$ ) be bijective.

Thus, if one calls a category structure on sets S and F the data of a quiver structure on these sets and of an associative composition law for which there exists at each vertex an identity element, one can take the functors as morphisms of the category structure (E, IV, p. 11).