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.