Since by axioms, there is always the identity morphism between the same object, we can express the above as condition on the cardinality of the hom-set
Some authors prefer a weaker notion, where a discrete category merely needs to be equivalent to such a category.
Any class of objects defines a discrete category when augmented with identity maps.
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. Alternately, for a general category C and the discrete category 2, one can consider the functor category C2. The diagrams of 2 in this category are pairs of objects, and the limit of the diagram is the product.
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. (For the right adjoint, see indiscrete category.)