Let $(F,G,\eta,\epsilon)$ be an equivalence of categories. Here $$F:\mathscr A\to \mathscr B,\ G:\mathscr B\to\mathscr A$$ are functors and $$\eta:1_\mathscr A\to GF,\ \epsilon:FG\to1_\mathscr B$$ are natural isomorphisms. Prove that $F$ is left adjoint to $G$.
This exercise is given (in Leinster's textbook) in the section about adjunction via initial objects, so I was thinking about applying the main result from that section, namely:
Is applying this theorem (i.e., proving that $\eta_A$ is initial in $(A\implies G)$ for all $A$) the right strategy?
But Leinster also pointed out before that if we have an equivalence $(F,G,\eta,\epsilon)$, then $\eta$ need not be the unit of adjunction. So if Theorem 2.3.6 implies that the $\eta$ from (b) must be the unit of adjunction (which is not stated, at least explicitly), then my strategy wouldn't work, right?
So my questions are whether I can/should use Theorem 2.3.6 in proving the stated result, and if not, then should I be using definitions only?
It is stated explicitly, in the proof of Thm. 2.3.6, that the morphism $\eta$, which has the property that for all $a\in\mathcal A$ the morphism $\eta_a$ is initial in $a\Rightarrow Ga$, is used as the unit for an adjunction.
However, this does not contradict that $(F,G,\eta,\varepsilon)$ might not be an adjunction: If $(F,G,\eta,\varepsilon)$ is an equivalence, then there is an adjoint equivalence $(F,G,\eta,\varepsilon')$ for some natural transformation $\varepsilon'$.
I haven't worked through the proof you refer to, but rather through the one in ncatlab, Thm. 3.3.
You construct $\varepsilon'$ as
$$\varepsilon'\colon FG\xrightarrow{\varepsilon^{-1}FG} FGFG\xrightarrow{F\eta^{-1}} FG\xrightarrow{\varepsilon} 1.$$
As a string diagram (where the transformations go up and morphisms are applied in right to left order), $\varepsilon'$ is
Now we can calculate
$$\begin{align} &\phantom{{}={}} \varepsilon F\circ F\eta^{-1}GF\circ \varepsilon^{-1}FGF\circ F\eta\\ &= \varepsilon F\circ F\eta\circ F\eta^{-1}\circ \varepsilon^{-1} F\\ &= \varepsilon F \circ \varepsilon^{-1}F\\ &= F\end{align}$$
See the linked article for a string diagram. The first equality holds since $\eta^{-1}\varepsilon^{-1}GF$ is 'distant' from $F\eta$ so that they commute. The other triangle identity follows from the lemma on ncatlab.