Preorder Category

Suppose Category is such that for any objects and there is at most one morphism Then is a binary relation on which is reflective and transitive and is called a preorder. A partially ordered set is a small preorder whose only isomorphisms are identities

Lemma

Every Morphism is a Monomorphism and Epimorphism. In particular, a preorder is balanced if and only if it is a Groupoid.