Let be a 2-category.
K(X,A) \to K(X,B)
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.
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 -fully-faithful functors 2-categorically; see codiscrete cofibration.