With the goal of approximating a semigroup by its largest factorisable subsemigroup, we define right \(\mathcal {U}\) -fair semigroups as the largest class of semigroups among which a naive approach makes sense. We study what it would take to extend the Morita theory of factorisable semigroups to the right \(\mathcal {U}\) -fair case. We show that there exists a coreflector functor from the category of acts into the category of firm acts and tie colimit preservation properties of this functor to the right \(\mathcal {U}\) -fairness of the semigroup. As the main result we show that being right \(\mathcal {U}\) -fair for a semigroup is equivalent to the coreflector preserving epimorphisms. We use this fact to prove some statements related to projective objects and Morita theory.
