A very elementary result in category theory allow us to promote equivalences of categories to adjoint equivalences by changing one of the two natural isomorphisms. Generalizing this, consider the following situation.
Let $F\colon\mathcal{A}\to\mathcal{B}$ and
$G\colon\mathcal{B}\to\mathcal{A}$ be two functors equipped with two natural
transformations $\eta\colon \text{id}_{\mathcal{A}}\Rightarrow GF$ and
$\epsilon\colon FG\Rightarrow\text{id}_B$ such that $\epsilon F\cdot F\eta$ and $G\epsilon\cdot\eta G$ are isomorphisms (i.e. we replace each identity of the triangle identities with an isomorphism).
I'd like to show (if it is true) that we can promote $F$ and $G$ to an adjoint pair by suitably replacing $\eta$ and $\epsilon$, ideally with an argument that holds in any $2$-category rather than just in $\mathbf{Cat}$.
Best Answer
As with the version with equivalences, you can choose to keep either $\eta$ or $\epsilon$. I will keep $\epsilon$.
First, we have the following identity: $$\begin{align*} G (\epsilon F \bullet F \eta)^{-1} \bullet \eta & = (G \epsilon \bullet \eta G)^{-1} F \bullet (G \epsilon \bullet \eta G) F \bullet G (\epsilon F \bullet F \eta)^{-1} \bullet \eta \\ & = (G \epsilon \bullet \eta G)^{-1} F \bullet G \epsilon F \bullet G F G (\epsilon F \bullet F \eta)^{-1} \bullet \eta G F \bullet \eta \\ & = (G \epsilon \bullet \eta G)^{-1} F \bullet G (\epsilon F \bullet F \eta)^{-1} \bullet G \epsilon F \bullet \eta G F \bullet \eta \\ & = (G \epsilon \bullet \eta G)^{-1} F \bullet G (\epsilon F \bullet F \eta)^{-1} \bullet G (\epsilon F \bullet F \eta) \bullet \eta \\ & = (G \epsilon \bullet \eta G)^{-1} F \bullet \eta \end{align*}$$
Let $\tilde{\eta} = G (\epsilon F \bullet F \eta)^{-1} \bullet \eta = (G \epsilon \bullet \eta G)^{-1} F \bullet \eta$. Then $\tilde{\eta}$ and $\epsilon$ satisfy the triangle identities: $$\begin{align*} \epsilon F \bullet F \tilde{\eta} & = \epsilon F \bullet F G (\epsilon F \bullet F \eta)^{-1} \bullet F \eta \\ & = (\epsilon F \bullet F \eta)^{-1} \bullet (\epsilon F \bullet F \eta) \\ & = \mathrm{id}_F \\ G \epsilon \bullet \tilde{\eta} G & = G \epsilon \bullet (G \epsilon \bullet \eta G)^{-1} F G \bullet \eta G \\ & = (G \epsilon \bullet \eta G)^{-1} \bullet (G \epsilon \eta G) \\ & = \mathrm{id}_G \end{align*}$$
To me, this is a thoroughly unenlightening proof. I think it is better to prove the special case of categories using the hom-set definition of adjunction and then "use Yoneda" to obtain the above proof for general 2-categories. It really comes down to the following observation: given composable maps $a, b, c$ such that $c \circ b$ and $b \circ a$ are invertible, $a, b, c$ are all invertible; in particular, $(c \circ b)^{-1} \circ c = a \circ (b \circ a)^{-1}$.