When is there an adjunction between two categories? Call two categories $C$ and $D$ adjivalent if there is an adjunction
$$F\dashv U,\quad F\colon C\to D,\quad U\colon D\to C$$
between $C$ and $D$.
Can one say something about when two categories are adjivalent? In particular, are there pairs of categories which aren't adjivalent?
 A: It's very common that categories should admit an adjunction between them. In particular if $C$ has a terminal and $D$ an initial object, then take $F$ constant at the initial and $G$ constant at the terminal. On the other hand, any adjunction between $C$ and $D$ induces a homotopy equivalence between their nerves, which means that "adjivalent" categories are weakly equivalent in the Thomason model structure on categories, though the converse need not hold.
A: If $C$ and $D$ are groupoids then an adjunction must be an equivalence (because the unit and the counit must be invertible), so groupoids have an adjunction between them iff they're equivalent. In particular one-object groupoids $BG, BH$ corresponding to two non-isomorphic groups $G, H$ do not have an adjunction between them.
By the way, I don't recommend calling this condition "-valent"; to me that suffix suggests an equivalence relation, which this isn't. For example, any category $C$ admits a unique functor $C \to 1$ to the terminal category, which has a left adjoint iff $C$ has an initial object and a right adjoint iff $C$ has a terminal object. And there are categories with an initial object but not a terminal object and vice versa. So "adjivalence" is not symmetric; it has a "handedness" depending on which direction you ask for a left adjoint and which direction you ask for a right adjoint.
