2-natural transformation?
A morphism $f\colon A\to B$ in a 2-category $K$ is said to be (representably) conservative if for all objects $X$, the induced functor
is conservative. In Cat, this is equivalent to $f$ being conservative in the usual sense.
Conservative morphisms often form the right class of a factorization system. In Cat, the left class consists of (possibly transfinitely) iterated localizations.
Of course, any fully faithful morphism is also conservative, and in particular any inverter or equifier is conservative. Moreover, any inserter is also conservative (and faithful), though not generally fully-faithful.
If $f$ is conservative in $K^{op}$ (the 1-cell dual), then $f$ is said to be liberal. This joke is due to Carboni, Johnson, Street, and Verity; see codiscrete cofibration.