Particularly, if $Y$ is dense, than $G$ is automaticallycanonically a pointwise Kan extension --- from density we have:
$$G(-) \approx \int^{A\in\mathbb{A}} \hom(Y(A), G(-)) \times Y(A)$$
and using the formula for an absolute lifting:
$$G(-) \approx \int^{A\in\mathbb{A}} \hom(F(A), -) \times Y(A)$$
The essence of the above example is that because the Yoneda functor $y_\mathbb{B} \colon \mathbb{B}\rightarrow \mathbf{Set}^{\mathbb{B}^{op}}$ is a full and faithful embedding, functors $F\colon\mathbb{A} \rightarrow \mathbb{B}$ may be thought as of distributors
$$y_\mathbb{B} \circ F = \hom(=, F(-))$$
Every distributor arisen in this way has a right adjoint distributor $\hom(F(=), -)$ in the bicategory of distributors. The distributor $\hom(F(=), -)$ has actually the type $\mathbb{B} \rightarrow \mathbf{Set}^{\mathbb{A}^{op}}$, which is the only think that may prevent $F$ of having the ordinary (functorial) right adjoint $G \colon \mathbb{B} \rightarrow \mathbb{A}$. Recall --- just recall, that we say that $F$ has a right adjoint, if there exists $G$ such that:
$$y_\mathbb{A} \circ G \approx \hom(F(=), -)$$
which means:
$$\hom(=, G(-)) \approx \hom(F(=), -)$$
Example: [2-topospowers from Yoneda traingle]triangle]
The motivating example is to start with a 2-functor $J \colon \mathbb{w} \rightarrow \mathbb{R}$$J \colon \mathbb{w} \rightarrow \mathbb{D}$ equipping a 2-category $\mathbb{w}$$\mathbb{W}$ with proarrows, and an extension $Y \colon \mathbb{w} \rightarrow \mathbb{W}$$Y \colon \mathbb{w} \rightarrow \overline{\mathbb{W}}$ embedding "small objects" into "locally small" (or large) objects in $\mathbb{W}$$\overline{\mathbb{W}}$. For simplicityThen to extend these data to the Yoneda triangle, we may think ofhave to find a functor $\mathbb{w}$$P \colon \mathbb{D} \rightarrow \overline{\mathbb{W}}$ representing a proarrow $A \nrightarrow B$ as a morphism $A \rightarrow P(B)$ in $\overline{\mathbb{W}}$, and a natural transformation $\eta \colon Y \rightarrow P\circ J$ playing the role of a familly of Yoneda morphisms $\eta_A \colon A \rightarrow P(A)$.
The archetypical situation is when we take $\mathbb{W} = \mathbf{cat}$, $\overline{\mathbb{W}} = \mathbf{Cat}$, $\mathbb{D} = \mathbf{Dist}$, where $\mathbf{cat}$ is the 2-category of small categories, $\mathbb{W}$ as of$\mathbf{Cat}$ is the 2-category of locally small categories, and $\mathbb{R}$ as of$\mathbf{Dist}$ is the bicategory of distributors between small categories. Then $J \colon \mathbf{cat} \rightarrow \mathbf{Dist}$, $Y \colon \mathbf{cat} \rightarrow \mathbb{Cat}$ are the Yoneda triangle on these data consistsusual embeddings, (additionally) of a functor$P \colon \mathbf{Dist} \rightarrow \mathbf{Cat}$ is the covarinat 2-power pseudofunctor $P \colon \mathbb{R} \rightarrow \mathbb{W}$ representing a proarrow$\mathbf{Set}^{(-)^{op}}$ defined on distributors via left Kan extensions, and $A \nrightarrow B$ as$\eta_\mathbb{A} \colon \mathbb{A} \rightarrow \mathbf{Set}^{\mathbb{A}^{op}}$ is the Yoneda embedding of a morphism $A \rightarrow P(B)$ insmall category $\mathbb{W}$$\mathbb{A}$.
In our example, this gives a usual representationWe know that there are isomorphisms of a distributor between small categories:
$$\hom_{\mathbf{Dist}}(\mathbb{A}, \mathbb{B}) \approx \hom_{\mathbf{Cat}}(\mathbb{A}, \mathbf{Set}^{\mathbb{B}^{op}})$$
where $\mathbb{A} \nrightarrow \mathbb{B}$ as$\mathbb{A}$ and $\mathbb{B}$ are small. Therefore, to show that $P$ is a functor $\mathbb{A} \rightarrow \mathbf{Set}^{\mathbb{B}^{op}}$(bi)pointwise left Kan extension it suffices to show that $Y$ is 2-dense. And the natural transformationHowever, $\eta \colon Y \rightarrow P \circ J$ from$Y$ is obviously 2-dense, because the Yoneda triangle consiststhe terminal category is a 2-dense subcategory of usual Yoneda embeddings $\mathbb{A} \rightarrow \mathbf{Set}^{\mathbb{A}^{op}}$$\mathbf{Cat}$ and $Y$ is fully faithful.
The point is that in most situations $\mathbb{R}$$\mathbb{D}$ is a monoidal bi-(bi)category, where the monoidal structure is inherited from a cartesianthe closed structure on $\mathbb{w}$$\mathbb{W}$. Moreover, functors and the natural transformation constituting the Yoneda triangle are (lax)monoidal. This means that monoids in $\mathbb{w}$$\mathbb{W}$ are mapped to the (pro)monoids in $\mathbb{R}$$\mathbb{D}$ which are mapped to monoids in $\mathbb{W}$$\overline{\mathbb{W}}$. If I am not mistaken this observation leads to an abstract characterisation of the concept of the Day convolution (and in a similar manner one may try to define a Dedekind-MacNeille completion of an object).
In our archetypical situation, a category $\mathbb{A} \times \mathbb{B}$ is mapped by $P$ to $\mathbf{Set}^{\mathbb{A}^{op} \times \mathbb{B}^{op}}$ and the missing morphisms making the unit of the triangle lax monoidal:
$$\mathbf{Set}^{\mathbb{A}^{op}} \times \mathbf{Set}^{\mathbb{B}^{op}} \rightarrow \mathbf{Set}^{\mathbb{A}^{op} \times \mathbb{B}^{op}}$$
is given by the convolution of the distributional identity $\mathbb{A} \times \mathbb{B} \nrightarrow \mathbb{A} \times \mathbb{B}$:
$$\langle F, G \rangle \mapsto \int^{A \in \mathbb{A}, B \in \mathbb{B}} F(A) \times G(B) \times \hom(-, A) \times \hom(=, B) = F(-) \times G(=)$$
Now, a promonoidal category $M \colon \mathbb{A} \times \mathbb{A} \nrightarrow \mathbb{A}$ is mapped by $P$ to:
$$H \mapsto \int^{\langle A, B \rangle \in \mathbb{A}\times \mathbb{A}} H(A, B) \times M(-, A, B)$$
and by composing it with the above map:
$$\langle F, G \rangle \mapsto \int^{\langle A, B \rangle \in \mathbb{A}\times \mathbb{A}} F(A) \times G(B) \times M(-, A, B)$$
we obtain the well-known formula for convolution.
