2-natural transformation?
Let $K$ be a 2-category.
A morphism $f:A\to B$ in $K$ is called (representably) fully-faithful (or sometimes just ff) if for all objects $X \in K$ , the functor
Fully faithful morphisms in a 2-category may also be called 1-monic, and be said to make their source into a 1-subobject of their target. See subcategory for some discussion.
Fully faithful morphisms are often the right class of a factorization system. The left class in $Cat$ consists of essentially surjective functors; in a regular 2-category it consists of the eso morphisms.
Just as in a 1-category any equalizer is monic, in a 2-category any inverter or equifier is fully faithful.
This is not always the “right” notion of fully-faithfulness in a 2-category. In particular, in enriched category theory this definition does not recapture the correct notion of enriched fully-faithfulness. It is possible, however, to characterize $V$-fully-faithful functors 2-categorically; see codiscrete cofibration.
In the 2-category Cat the full and faithful morphisms are precisely the full and faithful functors.