Let $C$, $D$ be categories and $F:C\to D$, $G: D\to C$ be functors.
Consider the following properties :
- $p_1$ : "$F$ is a left adjoint of $G$"
- $p_2$ : "$F$ has a right adjoint"
- $q_1$ : "$G$ is a right adjoint of $F$"
- $q_2$ : "$G$ has a left adjoint"
If you were to choose new names for these properties, what would be your choice ? Furthermore, what would be your notation for these properties?
PS: I'm asking this question because I don't like the usual terminology and I'm looking for a better one.
You can say "left / right dual" instead of left / right adjoint, and also "left / right dualizable" for the conditions that such adjoints exist. This terminology comes from the theory of dualizable objects in monoidal categories, which turn out to be essentially a specialization of left / right adjoints (once the latter are generalized to arbitrary 2-categories). "Dual" is unfortunately a very overloaded word in mathematics so it's not clear that this is really better.
One of the nice things about "adjoint" is that one of the only other uses of this word in mathematics is for a situation which is very analogous, namely adjoint linear transformations between inner product spaces. The idea is that there's an analogy between the hom-set definition of an adjunction
$$\text{Hom}(F(x), y) \cong \text{Hom}(x, G(y))$$
and the definition of an adjoint pair of linear transformations
$$\langle T(x), y \rangle = \langle x, T^{\dagger}(y) \rangle.$$
This analogy becomes very strong when considering Frobenius reciprocity, where the dimension of a hom space exactly recovers the inner product on characters of a finite or compact group.