TITLE: Kan extensions and the yoneda embedding.
QUESTION [6 upvotes]: [Edit: For a category $C$ let $C^\wedge$ denote the category of presheaves on $C$.]
Let $f:C\to D$ be a functor. Precomposition with $f^{op}$ induces a functor
$f^\wedge:D^\wedge \to C^\wedge$. This functor has both a left- and a right adjoint, called  left- and right kan extension:
$f_\wedge \dashv f^\wedge \dashv f_+$.
Now for $c\in C$ we get $D^\wedge(D(-,fc),Y)=Y(fc)=f^\wedge Y(c)=C^\wedge(C(-,c),f^\wedge Y)$. This gives us the restriction of $f_\wedge$ to $C$ along the yoneda embedding: It is $f$ (composed with the yoneda embedding).
Now here's my question:

What is the restriction of $f_+$ to $C$ along the yoneda embedding?

It seems not to agree with $f$ but:

Is there a nice connection between $f_+C(-,c)$ and $D(-,fc)$?

REPLY [4 votes]: I know its good "explanation".
Let $X, Y$ be sets, and $f:X\to Y$ be a map. Then, it induces a inverse image map $f^{-1}:P(Y)\to P(X)$. Regarding $P(X),P(Y)$ as categories by its inclusion order, $f^{-1}$ can be regard as a functor. Then, this functor has a left and right adjoint. The left adjoint functor is just an image $f_*:P(X)\to P(Y)$, and right adjoint $f_!:P(X)\to P(Y)$ is called "dual image, small image" defined by $f_!(U)=Y-f(X-U)$. It is easy to check they are actually adjoint.
This is a special case of Kan extensions. Because by taking $2=\{0\to 1\}$ and regarding $X$ as discrete category, we can describe $P(X)$ as just a functor category $2^X$. And the "composite with $f$ functor" is nothing but the inverse image map. 
Furthermore, these categories are $2$-enriched, so the contravariant Hom-functor $Hom(-,x):X^{op}=X\to 2$ is seen as an object of $2^X$. But except for $y=x$, $Hom(y,x)$ is empty. This means it's a "characteristic functor" of $\{x\}\in P(X)$. So, $\{-\}:X\to P(X), x\mapsto \{x\}$ is seen as a "$2$-enriched yoneda embedding".
In this case, the restriction of $f_*$ to X along the yoneda embedding is $f$, composed with the yoneda embedding. However, it is clear that there is no nice connection between $f_!$ and the yoneda embedding. I think this is a good analogy to explain the advantage of left Kan extension in combination with yoneda embedding.