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[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 $D(-,fc)$?
show/hide this revision's text 2 now the adjunctions are in math mode

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:

$f_\wedge\dashv f_\wedge \dashv f^\wedge \dashv f_+$.

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 $D(-,fc)$?
show/hide this revision's text 1

Kan extensions and the yoneda embedding.

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 $D(-,fc)$?