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
$$\eta' = ($\begin{array}{ccc} & 1_D & \stackrel{\eta}{\to} G H \ stackrel{G {}^{ \theta}{\to} G H').$$ 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,
