A smothering functor is a functor that is “almost an equivalence of categories” except that it may not be faithful. Smothering functors tend to arise when comparing “different homotopy categories” of the same category that impose more or less refined notions of homotopy equivalence. Frequently they can be treated more or less like equivalences.
If is smothering and satisfy in , then in , by fullness combined with conservativity. In other words, is “full on isomorphisms” (but since it is not faithful, it is not pseudomonic).
Further combining this with surjectivity on objects, every smothering functor is an isofibration.