Question

Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the appropriate (co)cone category, is there some (and if so, what is the) appropriate category for which, given a functor with a left/right adjoint, we can characterize said adjoint as its initial/terminal object? My goal in this is to characterize [initial objects, universals, limits, adjunctions] all as examples of each other.

Answer 1

Well, yes: the left adjoint of a functor $G: C \to D$ is the initial object in the category whose objects are pairs $(H: D \to C, \eta: 1_D \to G H)$ where $\eta$ is a natural transformation, and whose morphisms $(H, \eta) \to (H', \eta')$ are natural transformations $\theta: H \to H'$ such that

$$\begin{array}{ccc} & 1_D & \\ {}^{ \eta} \swarrow & & \searrow {}^{\eta'} \\ GH & \xrightarrow{G\theta} & GH'\end{array}$$

commutes. Similarly, a right adjoint can be expressed as a terminal object in a suitable category (exercise in applying the concept of duality).

See the discussion on comma categories in Categories for the Working Mathematician,