on the equivalence of notions of adjunctions let $C$ and $D$ be categories and $F : C \leftrightarrows D : G$ be a pair of functors. as is well known, there is a natural correspondence of

*

*pairs of inverse isomorphisms $α : \hom_D (FX, Y) \leftrightarrows \hom_C (X,GY) : β$ natural in $X$ and $Y$, and

*transformations $η \colon 1_C → GF$ and $ε \colon FG → 1_D$ such that $(εF)(Fη) = 1_F$ and $(Gε)(ηG) = 1_G$.

this correspondence essentially stems from the yoneda lemma, naturally in $X$ or $Y$ relating

*

*transformations $\hom_D(FX,–) → \hom_C(X,G–)$ to elements in $\hom_C(X,GFX)$, so to transformations $1_C → GF$,

*transformations $\hom_C(–,GY) → \hom_D(F–,Y)$ to elements in $\hom_D(FGY,Y)$, so to transformations $FG → 1_D$.

now, apparently, again by yoneda, the triangle identities for $1_F$ and $1_G$ relate to $βα = \mathrm {id}$ and $αβ = \mathrm {id}$ respectively.
– i’m having trouble seeing that!
let’s look at $βα = \mathrm {id}$: so $α$ and $β$ are morphisms of functors $C^\mathrm {op} × D → \mathrm {Set}$ with a composition
$$\hom_D (F–,–) → \hom_C(–,G–) → \hom_C(F–,–),$$
which by locally invoking the yoneda embedding for all $X$ in $C$ says that some morphism of functors $F → F$ is the identity. apparently this morphism is precisely $(εF)(Fη)$ in the correspondence? why is that?

If you have a natural (in both variables) transformation $\Phi:\mathbf D(F-,-)\to \mathbf C(-,G-)$, and hence an adjunction $F\dashv G:\mathbf D\to \mathbf C$, observe that any component of the unit $\eta_C:C\to GF(C)$ can be recovered as $\Phi(\operatorname{id}_{F(C)})$, and similarly any component of the counit $\varepsilon_D=\Phi^{-1}(\operatorname{id}_{G(D)})$.
Hence from one hand $\Phi^{-1}\circ\Phi (\operatorname{id}_{F(C)})=\operatorname{id}_{F(C)}$, as they are inverses. If however you start with $\operatorname{id}_{F(C)}$ and apply $\Phi$ you get $\eta _C$; applying $\Phi^{-1}$, you get exactly $\epsilon_{F(C)}\circ F(\eta_C)$, since the universal element of the natural transformation $\Phi^{-1}_{-,GF(C)}: \mathbf C(-,GF(C))\to \mathbf D(F-,F(C))$ is $\varepsilon_{F(C)}$; in fact, this last assertion is contained in the proof of Yoneda's lemma. Thus this shows that $\epsilon_{F(C)}\circ F(\eta_C)=\operatorname{id}_{F(C)}$; to obtain the  other triangular identity, you can reason dually.
