A supplied pointwise right adjoint extends uniquely to a functor

For $h:d\to d^{\mathrm{\prime }}$, postcomposition by $h$ gives a natural transformation $\mathcal{D}(F(-),d)\Rightarrow \mathcal{D}(F(-),d^{\mathrm{\prime }})$. Transport it through ${\theta }_d$ and ${\theta }_{d^{\mathrm{\prime }}}$; [F2] supplies a unique morphism $G(h):Gd\to G{d}^{\mathrm{\prime }}$ representing the result.

Postcomposition by an identity is the identity transformation, so Yoneda uniqueness gives $G(1_d)=1_{Gd}$.

Postcomposition by $kh$ is the composite of postcomposition by $h$ and by $k$, so Yoneda uniqueness gives $G(kh)=G(k)G(h)$. Thus $G$ is a functor.

The definition in step 1.1 makes $\theta$ natural in $d$; it was natural in $c$ by hypothesis. Therefore [L1] gives $F⊣G$.

If another functor structure made every ${\theta }_d$ natural, its value on $h$ would induce the same transported natural transformation, so [F2] would force it to equal $G(h)$. The supplied object assignment also shows that no class-sized selection was made in the proof.