Quiv (Rev #2) in nLab

DiGraph is the category of directed graphs. We can define a directed graph to be a functor G:X op→SetG : X^{op} \to Set where X opX^{op} is the category with an object 00, an object 11 and two morphisms s,t:1→0s,t : 1 \to 0, along with identity morphisms. This lets us efficiently define DiGraph as in CC as the category of presheaves on XX, where:

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

In other words, DiGraphDiGraph is the functor category from this X opX^{op} to Set.