I recently learned about two-variable adjunctions and multi-variable adjunctions, and about adjunctions in 2-categories. The nCatLab page on two-variable adjunctions talks about how to go from the Hom-set definition of adjunctions to the unit-counit definition. However, neither of these seems to mention the other definition of adjoints that I've seen, the definition in terms of universal morphisms. I've found this definition interesting because it connects limits and colimits to adjoint functors.

Is there a formulation of multi-variable adjunctions and/or adjunctions in 2-categories in terms of something like universal morphisms.

I think that, at least in the 2-category case, this might reduce to the following question: Is it possible to talk about initial/terminal "elements" of a comma object $f / g$ in $\mathcal C$, e.g., by talking about some property of morphisms from a terminal object 1 in $\mathcal C$? If so, when we replace comma categories with comma objects in the definition of adjoint functors in terms of universal morphisms (is this called "internalizing in $\mathcal C$?), is the resulting definition equivalent to the standard definition of adjunctions in 2-categories?