A morphism $f\colon A\to B$ in a 2-category $K$ is said to be (representably) faithful if for all objects $X$, the induced functor
is faithful. In Cat, this is equivalent to $f$ being faithful in the usual sense.
Faithful morphisms may also be called 2-monic and said to make their source into a 2-subobject of their target. See subcategory for discussion.
Faithful morphisms often form the right class of a factorization system. In Cat, the left class consists of essentially surjective and full functors.
Of course, any fully faithful morphism is also faithful, and in particular any inverter or equifier is faithful. Moreover, any inserter is also faithful, though not generally fully-faithful.