Summary: A transitive relational system is a pair \((A,R)\) where \(A \not = \emptyset\) and \(R\) is a transitive binary relation on \(A\). We define a congruence \(\theta\) on \((A,R)\) and a factor relation \(R / \theta\) on the factor set \(A / \theta\) such that the factor system \((A / \theta,R / \theta)\) is also a transitive relational system. We show that these congruences are in a one-to-one correspondence with the so-called LU-morphisms whenever the relation \(R\) is a quasiorder on \(A\).