A full and faithful functor is, surprisingly enough, a functor which is both full and faithful. “Full and faithful” is sometimes shortened to “fully faithful” or “ff.” See also subcategory.
Together with bijective-on-objects functors, ff functors form an orthogonal factorization system on . Less evilly, pair them with essentially surjective functors to get a bicategorial factorization system.
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