TITLE: Adjoints to the Evaluation functors
QUESTION [13 upvotes]: Let $C$ be a category. Consider $Psh(C)$ the category of presheaves on $C$. It is a well known fact that the evaluation functor at $c$ an object of $C$ of presheaves preserves both limits and colimits yet I can't find in the littérature nor compute right and left adjoints. If someone knows a formula for both adjoints it would be very helpful.

REPLY [2 votes]: Mac Lane has a good treatment of the Kan extensions in $CWM$. The following is a synopsis for the right Kan extension: 
given categories $A,M$ and $C$, the functor $K:M\to C$ induces a functor $A^{K}:A^{C}\to A^{M}$ defined by $A^KS=SK$ and $A^K(\sigma:S\overset{\cdot }{\rightarrow}S')=\sigma K:SK\overset{\cdot }{\rightarrow}S'K.$
In this situation, a right Kan extension is just a right adjoint to $A^K:$
given $T:M\to A,\ $ Ran$_TK$ will satisfy $\mathcal {Nat}[S,$Ran$_KT] \cong  \mathcal {Nat}[SK,T].$
In your problem, you have$M=1, C=C, A=\bf{Set}$ and $K:1 \to C$ which induces the functor eval: $\bf{Set}^{C} \to \bf{Set}^1$. 
The payoff of all this is that with a little work, you can exhibit Ran$_KT$ as a pointwise limit so these problems become computational. The details are in CWM (2d edition) on p. 234.
Similar results hold for the left Kan extension.