The category paths on a quiver. #
C is a quiver,
paths C is the category of paths.
When the quiver is itself a category #
path_composition : paths C ⥤ C.
We check that the quotient of the path category of a category by the canonical relation (paths are related if they compose to the same path) is equivalent to the original category.
Two functors out of a path category are equal when they agree on singleton paths.
A path in a category can be composed to a single morphism.