An adjunction \(\mathscr {L} \dashv \mathscr {R}\) means arrows \(\mathscr {L}(X) \to Y\) are essentially the same thing as arrows \(X \to \mathscr {R}(Y)\) for any \(X \in {\cal C}\) and \(Y \in {\cal D}\).
This means the diagram
commutes for any arrows \(f: \mathscr {L}(X) \to Y\) in \({\cal D}\).
The above can also be diagramatically denoted by transposition diagram
\[
\begin {array}{ccccccc}
X' & \xrightarrow {x} & X & \xrightarrow {\sigma _{X Y}(f)} & \mathscr {R}(Y) & \xrightarrow {\mathscr {R}(y)} & \mathscr {R}\left (Y'\right ) \\
\hline \mathscr {L}\left (X'\right ) & \xrightarrow {\mathscr {L}(x)} & \mathscr {L}(X) & \xrightarrow {f} & Y & \xrightarrow {y} & Y'
\end {array}
\]
or simply,
\[
\frac {X \to \mathscr {R}(Y) \quad ({\cal C})}{\mathscr {L}(X) \to Y \quad ({\cal D})}
\]
An adjunction is a concept that describes the relationship between two functors that are weakly inverse to each other [nakahira2023diagrammatic, sec. 4].
By "weakly inverse", we don't mean that applying one after the other gives the identity functor, but in a sense similar to eroding (i.e. enhancing holes) and dilating (i.e. filling holes) an image, applying them in different order yeilds upper/lower "bounds" of the original image [rosiak2022sheaf, sec. 7.1].
with unit \(\eta \) and counit \(\epsilon \), the diagrams
\[h=\eta _X \mathbin {\bullet } \mathscr {R}(h^{\sharp })\]
and
\[f^{\sharp }=\mathscr {L}(f) \mathbin {\bullet } \epsilon _Y\]
commute.