2-natural transformation?
A morphism $f\colon A\to B$ in a 2-category $K$ is said to be (representably) pseudomonic if for all objects $X$, the induced functor
is pseudomonic. In Cat, this is equivalent to $f$ being pseudomonic in the usual sense.
Pseudomonic morphisms may also be called (2,1)-monic and said to make their source into a (2,1)-subobject of their target. See subcategory for discussion.
Of course, any fully faithful morphism is also pseudomonic, and in particular any inverter or equifier is pseudomonic.