Here is one famouse approach of defining Adjoint functors:
We say $F: D\rightarrow C$ is left adjoint to $G:C \rightarrow D$ or equivalently $G$ is right adjoint to $F$ if $$ C\left(FY,X\right) \cong D\left(Y,GX\right) $$ is natural in both X and Y.
If I am right, an adjunction is such a system of $\left(C,D,F,G,\mu,\theta\right)$ where $\mu$ and $\theta$ are natural transformations or 2-morphism in the category of categories. Now my question is how the more general notion of adjointion in any 2-category can be formulated. As IAs I have learned from the answer bellow, two morphisms in a 2 category form an adjunction if they are dual to each other. I will be very thankful if one could give a link to some relevant readings or spell out the defenition of adjunction in a 2-category.