Usage of Yoneda lemma to show ${\rm Lan}_K \dashv \mathcal{E}^K$

Question

I am reading a proof showing that for $K:\mathcal{C}\rightarrow \mathcal{D}$ and $\mathcal{E}$ a category, the functor ${\rm Lan}_K$ is left adjoint to $\mathcal{E}^K$ and there is a thing I don't understand.

I was able to do it by showing the naturality of the bijective map $\phi_{-,?}: [\mathcal{D},\mathcal{E}]({\rm Lan}_K-,?)\rightarrow [\mathcal{C},\mathcal{E}](-,\mathcal{E}^K?)$ that maps $\gamma:{\rm Lan}_K F \rightarrow G$ to $\gamma_K\circ\eta^F$.

The author's argument is the following

Proof. By the Yoneda Lemma, any pair $(G,\gamma)$, as in the definition for the left Kan extension, yields a natural transformation $$\gamma^*:[\mathcal{D}.\mathcal{E}](G,-)\Rightarrow[\mathcal{C},\mathcal{E}](F,-\circ K)$$ by $\gamma_H^*(\alpha)=\alpha_K\circ\gamma$. The universal property of the left Kan extension says the natural transformation given by the pair $(\mathrm{Lan}_K F,\eta)$, $$\eta^*:[\mathcal{D},\mathcal{E}](\mathrm{Lan}_K F,-)\Rightarrow[\mathcal{C},\mathcal{E}](F,-\circ K)$$ is a natural isomorphism.

Which is equivalent to saying that the Yoneda lemma gives immediatly that my map $\phi$ is natural.

However I struggle to understand how this is related to the Yoneda lemma...

Answer 1

Let $C$ be any category which is locally small relative to $Set$. The Yoneda lemma tells you how a natural transformation $$\alpha:C(c,-)\Rightarrow F$$ out of a representable functor looks like. It says that any such transformation is uniquely determined by the element $\alpha_c(id_c)\in Fc$. Here $F$ must of course be a functor of the format $F:C\to Set$. Given an element $\xi\in Fc$, the transformation $\alpha$ associated to it is the one whose $x$th component is the function $\alpha_x:C(c,x)\to Fx$ for which $\alpha_x(u) = (Fu)(\xi)$.

The Yoneda lemma thus tells you that there is a bijection between transformations $\alpha:C(c,-)\Rightarrow F$ and elements $\xi\in Fc$. The explicit description of the bijection shows that the transformation $\alpha$ associated to $\xi\in Fc$ is an isomorphism if and only if the pair $(c,\xi)$ satisfies the following universal property: whenever $x$ is an object and $\zeta\in Fx$ is an element, then there is a unique map $u\in C(c,x)$ such that $(Fu)(\xi) =\zeta$.

Now use this in your example. The category $C$ in question is the functor category $\mathcal E^\mathcal D$ and the functor $F$ is the functor $\mathcal E^\mathcal C(F,-K):\mathcal E^\mathcal D\to Set$. The discussion above says that a natural isomorphism $$\mathcal E^\mathcal D(G,-)\Rightarrow\mathcal E^\mathcal C(F,-K)$$ is the exact same thing as an element $\gamma \in \mathcal E^\mathcal C(F,GK)$ such that the pair $(G,\gamma)$ satisfies the universal property described above. You can check that the universal property of $(G,\gamma)$ is exactly that of a left Kan extension.