Familiar background (partly to fix notation). Suppose we have functors $F\colon \mathscr{A} \to \mathscr{B}$, $G\colon \mathscr{B} \to \mathscr{A}$ such that $F \dashv G$, and functors $F'\colon \mathscr{B} \to \mathscr{C}$, $G'\colon \mathscr{C} \to \mathscr{B}$ such that $F' \dashv G'$. We want to prove the adjunctions compose, so $F'F \dashv GG'$.
One way is to do it via homsets.
Another is via units and counits. Suppose $\eta, \varepsilon$ are the unit and counit of the first adjunction, and $\eta', \varepsilon'$ the unit and counit of the second adjunction. Then we evidently have natural transformations $\eta'', \varepsilon''$ defined by composition as follows: $$\eta'': \quad 1_{\mathscr{A}}\overset{\eta}\Longrightarrow GF \overset{G\eta'F}\Longrightarrow GG'F'F$$ $$\varepsilon'': \quad F'FGG' \overset{F'\epsilon G'}\Longrightarrow F'G' \overset{\varepsilon'}\Longrightarrow 1_{\mathscr{C}}$$ So to complete the proof that $F'F \dashv GG'$ it "just" remains to show by a diagram chase that $\eta'', \varepsilon''$ are a unit and counit for this adjunction because they satisfy the triangle equalities.
Mac Lane in effect sets this as an exercise for the reader at the bottom of p. 103 of Categories for the Working Mathematician. But this reader seems to be having a senior moment (well, a few moments), which is why I am asking here the embarrassingly undergraduate ...
Question How does the diagram chase for one of the required triangle equalities actually go?