On the semigroup of monoid endomorphisms of the semigroup ${\mathcal{C}}_{+}(a,b)$

Let ${\mathcal{C}}_{+}(a,b)$ be the submonoid of the bicyclic monoid which is studied in \cite{Makanjuola-Umar=1997}. We describe monoid endomorphisms of the semigroup ${\mathcal{C}}_{+}(a,b)$ which are generated by the family of all congruences of the bicyclic monoid and all injective monoid endomorphisms of ${\mathcal{C}}_{+}(a,b)$.