Let $\mathcal {C,D:Cat}$ be locally small categories for simplicity, and $L:\mathcal{C\rightarrow D}$ and $R:\mathcal{D\rightarrow C}$ functors.
Let $F:\mathcal{C\times D}\rightarrow Set$ be a bifunctor with $F\ (c,d) := Hom_{\mathcal D}\ (L\ c, d)$,
$G:\mathcal{C\times D}\rightarrow Set$ also be a bifunctor with $G\ (c,d) := Hom_{\mathcal C}\ (c, R\ d)$
and $\alpha:F\cong G$ a (bi-)natural isomorphism between the two.
tldr; $L\dashv R$.
The question: I want to construct the unit and counit from this definition.
We can define in components
$\eta_c := \alpha_{(c,L\ c)}\ id\ :\ c\rightarrow(R\circ L)\ c$ and
$\varepsilon_d := \alpha^{-1}_{(R\ d, d)}\ id\ :\ (L\circ R)\ d\rightarrow d$,
and prove these are indeed natural transformations.
But how can I prove that $(\varepsilon\circ L)\ \circ\ (L\circ\eta) = id$ ?
(Let's focus on this identity since the other one is analogous.)
Thanks in advance!
The naturality of $\alpha$ lets us say
$$\require{AMScd} \begin{CD} Hom_D\ \big((L\circ R)\ d, d\big) @<{\alpha^{-1}_{(R\ d, d)}}<< Hom_C\ \big(R\ d, R\ d\big)\\ @V g\ \circ\_\circ\ L\ f VV @VV R\ g\ \circ\_\circ\ f V \\ Hom_D\ \big(L\ c, L\ c\big) @<<{\alpha^{-1}_{(c, L\ c)}}< Hom_C\ \big(c, (R\circ L)\ c\big) \end{CD}$$
for $f:c\rightarrow R\ d$ and $g:d\rightarrow L\ c$.
In particular if $d = L\ c$ we can use $f:=\eta_c$ and $g:=id_{L\ c}$ and we get $$ \varepsilon_{L\ c}\circ L\ \eta_c = g \circ \big(\alpha^{-1}_{((R\circ L)\ c,L\ c)}\ id_{R\ d}\big) \circ L\ f = \alpha^{-1}_{(c,L\ c)}\ \big(R\ g \circ id_{R\ d} \circ f\big) = id_{L\ c} $$