Since a string diagrams is composed from top to bottem, left to right, we can read
as
\[(X \mathbin {\bullet } \mathscr {F}) \mathbin {\bullet } (\alpha _X) \mathbin {\bullet } (f \mathbin {\bullet } \mathscr {G})=(X' \mathbin {\bullet } \mathscr {G}) \quad {\large =} \quad (X \mathbin {\bullet } \mathscr {F}) \mathbin {\bullet } (f \mathbin {\bullet } \mathscr {F}) \mathbin {\bullet } (\alpha '_X)=(X' \mathbin {\bullet } \mathscr {G})\]
where each pair of parentheses corresponds to an overlay in the string diagram,
or with the notation in the opposite direction that is more familiar to most:
\[\mathscr {G}(f) \circ \alpha _X \circ \mathscr {F}(X) = \mathscr {G}(X') \quad {\large =} \quad \alpha '_X \circ \mathscr {F}(f) \circ \mathscr {F}(X) = \mathscr {G}(X')\]
Note that we read the wire from \(\mathscr {F}\) to \(\mathscr {G}\) as \(\mathscr {F}\) before the natural transformation, but as \(\mathscr {G}\) after the transformation.