Under local smallness, transposition gives the natural hom-set bijection, and conversely

Given an adjunction, [F1] makes the displayed hom-collections sets, and [L1] shows that ${\mathrm{\Phi }}_{c,d}$ and ${\mathrm{\Psi }}_{c,d}$ are inverse bijections.

For $a:c^{\mathrm{\prime }}\to c$, $b:d\to d^{\mathrm{\prime }}$, and $u:Fc\to d$, functoriality and naturality of $\eta$ give ${\mathrm{\Phi }}_{c^{\mathrm{\prime }},d^{\mathrm{\prime }}}(b\circ u\circ F(a))=G(b)\circ {\mathrm{\Phi }}_{c,d}(u)\circ a$; this is naturality in both variables in the sense of [F2].

Conversely, let $\mathrm{\Phi }$ be a natural family of bijections with inverse $\mathrm{\Psi }$. Define ${\eta }_c:={\mathrm{\Phi }}_{c,Fc}(1_{Fc})$ and ${\epsilon }_d:={\mathrm{\Psi }}_{Gd,d}(1_{Gd})$.

Naturality of $\mathrm{\Phi }$ in $c$ makes $\eta$ natural, and naturality of $\mathrm{\Psi }$ in $d$ makes $\epsilon$ natural.

Naturality in $d$ and $c$, respectively, gives ${\mathrm{\Phi }}_{c,d}(u)=G(u)\circ {\eta }_c$ and ${\mathrm{\Psi }}_{c,d}(v)={\epsilon }_d\circ F(v)$.

Since $\mathrm{\Psi }\mathrm{\Phi }$ fixes $1_{Fc}$, step 2.2 gives ${\epsilon }_{Fc}\circ F({\eta }_c)=1_{Fc}$; since $\mathrm{\Phi }\mathrm{\Psi }$ fixes $1_{Gd}$, it gives $G({\epsilon }_d)\circ {\eta }_{Gd}=1_{Gd}$.

Thus the data of steps 1.3 and 2.1 satisfy both triangle identities and define an adjunction. Any unit and counit inducing $\mathrm{\Phi }$ must be the transposes of the two identity morphisms, so step 1.3 also proves uniqueness.