I think there's a confusion between two very separate and distinct scenarios, where similar names are used for different things.
> Consider three functions \\( L, C, R \\) where \\( C : A \rightarrow B \\) and \\( L, R : B \rightarrow A \\).
Where \\( L \\) and \\( R \\) are left and right adjoint to \\( C \\) respectively.
The core issue here might be the fact that \\(C : A \to B\\) is both a left adjoint and a right adjoint. We have _two_ adjunctions here: \\(L \dashv C\\) and \\(C \dashv R\\).
However, for an adjunction in general, we might name the two halves of the conjunction by \\(L \dashv R\\). In this scenario, there is no \\(C\\).