Any class of objects defines a discrete category when augmented with identity maps.
Also, a category is discrete if and only if all of its subcategories are full.
The limit of any functor from a discrete category into another category is called a product, while the colimit is called a coproduct.
Thus, for example, the discrete category with just two objects can be used as a diagram or diagonal functor to define a product or coproduct of two objects.
The functor from Set to Cat that sends a set to the corresponding discrete category is left adjoint to the functor sending a small category to its set of objects.