Recall in a $2$-category $X$, a $1$-cell $f:X\to Y$ is called an equivalence provided there exists a $1$-cell $g:Y\to X$ together with the data of a pair of isomorphisms $\eta\_X: gf \to \operatorname{id}\_X$ and $\eta\_Y: fg \to \operatorname{id}\_Y$. 


Then suppose we impose the additional requirement that $f\ast \eta\_X = \eta\_Y \ast f$ and $g\ast \eta\_Y = \eta\_X \ast g$. 


Does this idea have a name? Is it the case that for any equivalence, we can find a pair $\eta\_X, \eta\_Y$ for which this holds?



Is there a name for this type of equivalence in a 2-category?
This is well-known under the name [adjoint equivalence](http://ncatlab.org/nlab/show/adjoint+equivalence) (when you replace $\eta\_Y$ by $\eta\_Y^{-1}$ in your data). And yes, every equivalence can be modified to some adjoint equivalence (one may fix the unit, but has to modify the counit; or vice versa).


