nLab
Quiv

$\mathrm{Quiv}$ or $\mathrm{DiGraph}$ is the category of quivers or (as category theorists often call them) directed graph.

We can define a quiver to be a functor $G:X^{\mathrm{op}}\to \mathrm{Set}$, where $X^{\mathrm{op}}$ is the category with an object $0$, an object $1$ and two morphisms $s,t:1\to 0$, along with identity morphisms. This lets us efficiently define $\mathrm{Quiv}$ as the category of presheaves on $X$, where:

• objects are functors $G:X^{\mathrm{op}}\to C$,
• morphisms are natural transformations between such functors.

In other words, $\mathrm{Quiv}$ is the functor category from this $X^{\mathrm{op}}$ to Set.

category: category