The presheaf category on a small category is the free cocompletion

Fix a presheaf $P$. Under the Yoneda bijection [L4], an object of the comma category $(y\downarrow P)$ is exactly a pair $(c,x)$ with $x\in P(c)$, and the comma-category morphism condition says that a map $(c,x)\to (d,y)$ is precisely an arrow $f:c\to d$ with $x=P(f)(y)$, exactly as in the category of elements (The category of elements of a covariant functor or a presheaf, The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding). So $(y\downarrow P)$ is canonically $\int P$, the same indexing category that appears in the self-Kan presentation [L1], and the displayed diagram is the comma-category diagram used by [L6]. Therefore its chosen colimit is the pointwise left Kan extension value of $F$ along $y$ at $P$. Because these colimits and their universal cocones are supplied for every $P$, [L6] assembles them into a functor ${\mathrm{Lan}}_{y}F:[{\mathcal{C}}^{\mathrm{o}\mathrm{p}},\mathbf{S}\mathbf{e}\mathbf{t}]\to \mathcal{E}$. By [L7] the functor $y$ is fully faithful, so [L2] shows that this pointwise extension restricts along $y$ to $F$ up to the canonical isomorphism.

Let $e\in \mathcal{E}$, and define the presheaf $Q_e:=\mathcal{E}(F-,e)$ on $\mathcal{C}$. A morphism ${\mathrm{Lan}}_{y}F(P)\to e$ is, by the chosen colimit universal property, exactly a cocone from the diagram to the constant diagram at $e$. By [L4], giving such a cocone is equivalently giving, for each object $(c,x)$ of $\int P$, an element of $Q_e(c)$ natural in morphisms of $\int P$, that is, a cocone from the canonical density diagram of $P$ to $Q_e$. By [L3], these cocones are exactly natural transformations $P\Rightarrow Q_e=P\Rightarrow \mathcal{E}(F-,e)$. Hence $\mathcal{E}({\mathrm{Lan}}_{y}F(P),e)\cong [{\mathcal{C}}^{\mathrm{o}\mathrm{p}},\mathbf{S}\mathbf{e}\mathbf{t}](P,\mathcal{E}(F-,-))$ naturally in $P$ and $e$, so ${\mathrm{Lan}}_{y}F⊣\mathcal{E}(F-,-)$.

Now let $L:[{\mathcal{C}}^{\mathrm{o}\mathrm{p}},\mathbf{S}\mathbf{e}\mathbf{t}]\to \mathcal{E}$ preserve all small colimits and put $F:=Ly$. By [L3], every presheaf $P$ is the colimit of its canonical density diagram of representables. Applying $L$ preserves that colimit, so $L(P)$ is the colimit of the diagram $(c,x)\mapsto Ly(c)=F(c)$. By step 1.1 this is exactly ${\mathrm{Lan}}_{y}F(P)$, naturally in $P$. Hence $L\cong {\mathrm{Lan}}_y(Ly)$.

Since ${\mathrm{Lan}}_{y}F$ is a left adjoint by step 2.1, [L8] shows that it preserves all small colimits.

Conversely, let $L⊣H$ with domain the presheaf category, and put $F:=Ly$. For any $e\in \mathcal{E}$ and $c\in \mathcal{C}$, adjunction and [L4] give $H(e)(c)\cong [{\mathcal{C}}^{\mathrm{o}\mathrm{p}},\mathbf{S}\mathbf{e}\mathbf{t}](y(c),H(e))\cong \mathcal{E}(Ly(c),e)=\mathcal{E}(F(c),e)$, naturally in $c$ and $e$. So $H\cong \mathcal{E}(F-,-)$, and step 2.1 shows ${\mathrm{Lan}}_{y}F$ is another left adjoint to the same right adjoint. Therefore [L5] gives $L\cong {\mathrm{Lan}}_{y}F$.

Given $\tau :F\Rightarrow F^{\mathrm{\prime }}$, its components induce a morphism between the two chosen density colimits at every presheaf, and the colimit universal properties make these morphisms natural. Conversely, any natural transformation between small-colimit-preserving extensions is determined at $P$ by its components on the representables in the canonical density colimit of [L3]. Therefore extension of $\tau$ is unique, which completes the free-cocompletion universal property.