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\$$.