Do generalizations of adjoint functors, such as adjunctions in 2-categories, and multivariable adjunctions, have formulations in terms of something like universal morphisms? 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?
 A: The “universal morphism” formulation is easily adapted to multi-variable adjunctions though a little tedious to describe.
If you fix any $F : C\times D \to E$, the adjoint $G : D^o \times E \to C$ is specified by the fact that for each $d, e$ there is a universal morphism $\epsilon : F(G(d,e), d) \to e$ in that any other $h : F(c,d) \to e$ there exists a unique $\hat h : c \to G(d,e)$ that factorizes as $h = \epsilon \circ F(\hat h, id_d)$. This is probably most familiar from the definition of an internal hom in terms of the universality of the application map. But note that you can also specify $G$ from the other adjoint $H : C^o \times E \to D$ by a universal morphism $d \to H(G(d,e),e)$ or you could specify $F$ by a universal morphism $\eta : c \to G(d,F(c,d))$ or $d \to H(c,F(c,d))$.
To me the clearest way to describe the universal morphism construction is in terms of Representable functors. All of the above universal morphisms are instances of the following situation. You are given a pro-functor $R : C^o \times D \to Set$ and you seek a functor $G : D \to C$ with a natural isomorphism $c \to G(d) \cong R(c,d)$. This is determined by, for each $c$, a universal element $\epsilon : R(G(d),d)$, where universality means for any other $r : R(c, d)$ there exists a unique $\hat r : c \to G(d)$ that factorizes $R(\hat r, id_d)(\epsilon) = r$. The above description of multi-variable adjoints in these terms can all be systematically derived from this lemma by picking the right profunctor $R$.
Now, on the second question of internalizing this definition in a 2-category, it cannot be completely internalized in an arbitrary 2-category, because part of the definition in terms of universal morphisms is that $G$ above is a functor but the universal morphism description allows us to define the functor $G$ by giving an action on objects only, because the action of $G$ on morphisms can be derived from the universal morphism $G(f : d \to d’) = \hat r$ where $r = {R(id,f)(\epsilon)}$. In a 2-category we don’t have a way to talk about “functions on objects” like this as opposed to a 1-cell.
But with this caveat, the universal morphism description can be internalized in a 2-category-like structure: a pro-arrow equipment or even better a virtual equipment is a structure based on a double category, which is like a 2-category but where there are two kinds of 1-cells: in this case, think functors $F : C \to D$ and profunctors $R : C^o \times D \to Set$. Then in a virtual equipment you can prove that the following data are equivalent:

*

*An isomorphism of pro-arrows $R(c,d) \cong C(c,G(d))$

*2-cells $\epsilon : \forall d. R(G(d),d)$ and $\hat\cdot : R(c,d) \to C(c,G(d)$ satisfying a factorization property.

And you can also show that $G$’s action on morphisms in such a case must be equivalent to the definition in terms of the universal morphism. This still doesn’t quite reproduce the original equivalence because if you unravel this definition in the equipment of categories you will have a lot of naturality conditions that are not present in the original formulation, but this is because in the formal category theory setting you can’t directly talk about “functions on objects” or “possibly unnatural transformations”. However, you can get a generalization of the original by considering adjoints like this in a virtual equipment of the form $Mod(\mathcal V)$, because these are “categories internal to $\mathcal V$” and so functors are given by 1-cells in $\mathcal V$ with a 2-cell giving the action on morphisms and 2-cells come with a naturality condition in $\mathcal V$, etc.
