TITLE: Composition of adjunctions in a (weak) 2-category
QUESTION [5 upvotes]: Let $(f_1, g_1, \varepsilon_1, \eta_1)$ and $(f_2, g_2, \varepsilon_2, \eta_2)$ be adjunctions in a (weak) 2-category. Then there is an adjunction $(f_2 \circ f1, g1 \circ g2, \varepsilon, \eta)$. I can figure out how to define $\varepsilon$ (resp., $\eta$) by inserting appropriately the unitors and associators between $\varepsilon_1$ and $\varepsilon_2$ (resp., $\eta_1$ and $\eta_2$).
Although I know how to prove the counit-unit equations in the special case of the 2-category of categories (See for instance this answer), I cannot figure out how to prove them in the general case.
Could someone spell out the commutative diagrams or provide with a reference where it is done in full?
It is stated in Section 2.1 of Review of the elements of 2-categories (Kelly and Street, 1974), but there is unfortunately no proof there.

REPLY [3 votes]: Analogously to the case of Sets in 1-category theory, it’s often enough to prove a statement for categories and apply the Yoneda embedding. This maps an adjunction in $K$ to an adjunction in the 2-category of 2-functors $K^{op}\to Cat$. Since the 2-Yoneda embedding is 2-fully faithful, it’s enough to prove the triangle identities for the composition in the latter 2-category. And here the definition is given levelwise by the definition in Cat, so if you can prove the triangle identities in Cat, you’ve proved them in $K$!