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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?

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This is well-known under the name 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).

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