Let the following commutative diagram of functors be given: Open image in new window Then a general criterion is proved which allows to show the existence of U-semifinal liftings provided V admits the formation of certain V-semifinal liftings. Several results appear as corollaries, in particular sufficient (and partly necessary) conditions to get the implications V right adjoint ⇒ U right adjoint V semi-cofibration ⇒ U semi-cofibration V semi-topological ⇒ U semi-topological. Furthermore, a cancellability theorem and the following existence theorem for semi-topological functors can be derived from the general lifting theorem: U is semi-topological iff U is faithful, U preserves all limits and A contains certain (large) limits.