Initial morphisms and monomorphisms

Initial cones are shown to be mono-cones for several important types of functors, namely those possessing a reducible generator or having a certain topological (1.7) resp. algebraic property (1.8). These results are useful for the computation of the universal initial completion of faithful, topologically algebraic functors, contain results of Börger-Tholen and Hoffmann for certain setbased functors as special cases and allow a complete characterization of the initial cones in quite a few categories.