full and faithful functor



A full and faithful functor is a functor which is both full and faithful. “Full and faithful” is sometimes shortened to “fully faithful” or “ff.” See also full subcategory.


Together with bijective-on-objects functors, ff functors form an orthogonal factorization system on CatCat. Less evilly, pair them with essentially surjective functors to get a bicategorial factorization system.

Higher categorical generalizations

There is a bigger pattern at work here which is indicated at stuff, structure, property and k-surjective functor.

For (∞,1)-categories the corresponding notion of fully faithful functor is described at

Revised on May 11, 2012 12:05:30 by Urs Schreiber (